Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
prabau merged 6 commits into from
Feb 14, 2026
Merged

Give some spaces the Toronto trait #1621

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
6 commits
Select commit Hold shift + click to select a range
3138802
traits
felixpernegger Feb 7, 2026
6421519
Update spaces/S000017/properties/P000219.md
felixpernegger Feb 8, 2026
5ee3667
typo
felixpernegger Feb 8, 2026
0c2f03e
typo
felixpernegger Feb 8, 2026
974fcac
typo
felixpernegger Feb 8, 2026
23921b9
apply suggestions
felixpernegger Feb 13, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
7 changes: 7 additions & 0 deletions spaces/S000017/properties/P000219.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.
13 changes: 13 additions & 0 deletions spaces/S000199/properties/P000219.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.
13 changes: 13 additions & 0 deletions spaces/S000200/properties/P000219.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.
14 changes: 14 additions & 0 deletions spaces/S000217/properties/P000219.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.
, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
 blocks\n(function() {\n function addCopyButtons() {\n document.querySelectorAll('pre code').forEach(function(codeBlock) {\n if (codeBlock.parentElement.hasAttribute('data-copy-added')) return;\n codeBlock.parentElement.setAttribute('data-copy-added', 'true');\n \n var btn = document.createElement('button');\n btn.textContent = 'Copy';\n btn.style.cssText = 'position:absolute;top:4px;right:4px;padding:2px 8px;font-size:11px;background:#4ecdc4;border:none;border-radius:4px;color:#1a1a2e;cursor:pointer;opacity:0.7;transition:opacity 0.2s;';\n btn.onmouseover = function() { this.style.opacity = '1'; };\n btn.onmouseout = function() { this.style.opacity = '0.7'; };\n btn.onclick = function() {\n navigator.clipboard.writeText(codeBlock.textContent).then(function() {\n btn.textContent = 'Copied!';\n setTimeout(function() { btn.textContent = 'Copy'; }, 1500);\n });\n };\n codeBlock.parentElement.style.position = 'relative';\n codeBlock.parentElement.appendChild(btn);\n });\n }\n \n addCopyButtons();\n \n // Re-run on dynamic content\n var observer = new MutationObserver(addCopyButtons);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Add Copy Buttons to Code Blocks");
}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Universal Dark Mode - works on any site\n(function() {\n var enabled = true;\n \n function applyDarkMode() {\n if (!enabled) return;\n \n // Create style element if it doesn't exist\n var style = document.getElementById('universal-dark-mode-style');\n if (!style) {\n style = document.createElement('style');\n style.id = 'universal-dark-mode-style';\n document.head.appendChild(style);\n }\n \n // Dark mode CSS - inverts colors but preserves images/video\n style.textContent = '\n /* Invert everything except media */\n html {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #1a1a2e !important;\n }\n \n /* Restore images, videos, iframes, canvas */\n img, video, iframe, canvas, svg, picture, [style*=\"background-image\"] {\n filter: invert(1) hue-rotate(180deg) !important;\n }\n \n /* Preserve specific elements that should not be inverted */\n .no-dark-mode, .no-dark-mode *,\n [data-theme=\"light\"], [data-theme=\"light\"],\n .ace_editor, .ace_editor *,\n .CodeMirror, .CodeMirror *,\n .monaco-editor, .monaco-editor *,\n .markdown-body pre, .markdown-body pre *,\n .highlight, .highlight *,\n pre code, pre code * {\n filter: none !important;\n }\n \n /* Fix common UI elements */\n .modal, .popup, .dropdown-menu, .tooltip, .popover {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #2d2d44 !important;\n border-color: #444 !important;\n }\n \n /* Scrollbars */\n ::-webkit-scrollbar { background: #1a1a2e !important; }\n ::-webkit-scrollbar-thumb { background: #444 !important; }\n ::-webkit-scrollbar-thumb:hover { background: #555 !important; }\n \n /* Selection */\n ::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ';\n }\n \n function removeDarkMode() {\n var style = document.getElementById('universal-dark-mode-style');\n if (style) style.remove();\n }\n \n // Toggle with Alt+Shift+D\n document.addEventListener('keydown', function(e) {\n if (e.altKey && e.shiftKey && e.key === 'D') {\n e.preventDefault();\n enabled = !enabled;\n if (enabled) {\n applyDarkMode();\n console.log('[Universal Dark Mode] Enabled');\n } else {\n removeDarkMode();\n console.log('[Universal Dark Mode] Disabled');\n }\n }\n });\n \n // Apply on load\n applyDarkMode();\n \n // Re-apply on dynamic content\n var observer = new MutationObserver(function(mutations) {\n if (enabled && !document.getElementById('universal-dark-mode-style')) {\n applyDarkMode();\n }\n });\n observer.observe(document.head, { childList: true });\n \n console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');\n})();", "Universal Dark Mode");
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();

Skip to content
Give some spaces the Toronto trait#1621

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

 Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and
privacy statement. We’ll occasionally send you account related emails.

Already on GitHub?
Sign in
to your account

 Merged

prabau
merged 6 commits into

from

Feb 14, 2026
 Merged


Give some spaces the Toronto trait
#1621
Show file tree
Hide file tree
 Changes from all commits
Commits

Show all changes


6 commits


Select commit
Hold shift + click to select a range
3138802

traits


felixpernegger Feb 7, 2026
6421519

Update spaces/S000017/properties/P000219.md


felixpernegger Feb 8, 2026
5ee3667

typo


felixpernegger Feb 8, 2026
0c2f03e

typo


felixpernegger Feb 8, 2026
974fcac

typo


felixpernegger Feb 8, 2026
23921b9

apply suggestions


felixpernegger Feb 13, 2026

File filter

Filter by extension
Filter by extension
Conversations

Failed to load comments. 
Loading
Jump to

Jump to file


Failed to load files. 
Loading
 Diff view

 Diff view


7 changes: 7 additions & 0 deletions
spaces/S000017/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
---
space: S000017
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $f:Y\to X$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000199/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000199
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

13 changes: 13 additions & 0 deletions
spaces/S000200/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
space: S000200
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
There is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$[y,\to)_X\cap Y = [y,\to)_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.

14 changes: 14 additions & 0 deletions
spaces/S000217/properties/P000219.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,14 @@
---
space: S000217
property: P000219
value: true
---

Let $Y\subseteq X$ with $|Y|=|X|$.
Using well-orderedness and the fact that $\omega_1$ is an initial ordinal,
one can show that there is a unique order isomorphism $f:Y\to X$.
Since $X$ is Alexandrov, so is $Y$ with its subspace topology $\tau_Y$.
The smallest $\tau_Y$-neighborhood of a point $y\in Y$ is
$$(\leftarrow,y]_X\cap Y = (\leftarrow,y]_Y,$$
which coincides with the smallest neighborhood of $y$ in the left ray topology of $Y$.
This shows that $f$ is a homeomorphism.