Toronto + not hyperconnected + finite => has an isolated point - #1786

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@felixperneggerfelixpernegger commented May 23, 2026

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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.

@prabau

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The proof looks good.

Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?

@prabau

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?

@felixpernegger

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?

Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2
So this might just be a coincidence

@felixpernegger

felixpernegger commented May 24, 2026

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The proof looks good.

Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?

I found this to be the most natural way to state it, but if you want me to change it, sure just say

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Update theorems/T000898.md
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Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau merged commit 7796ddc into mainMay 24, 2026
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Toronto + not hyperconnected + finite => has an isolated point (#1786)
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, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
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})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

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Toronto + not hyperconnected + finite => has an isolated point - #1786
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Toronto + not hyperconnected + finite => has an isolated point#1786
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@felixpernegger
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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
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May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
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May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
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May 24, 2026
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Comment threadtheorems/T000898.md
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@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
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@prabau
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deleted the

torontoisolated

branch
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felixpernegger added a commit
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@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
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, '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");
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(function(){
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var __m = "*";
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Toronto + not hyperconnected + finite => has an isolated point - #1786
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Toronto + not hyperconnected + finite => has an isolated point#1786
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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
prabau
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May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
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May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
felixpernegger
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May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
prabau
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May 24, 2026
View reviewed changes
Comment threadtheorems/T000898.md
Outdated

@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
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@prabau
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deleted the

torontoisolated

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@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
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, '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");
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} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Toronto + not hyperconnected + finite => has an isolated point - #1786
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prabau merged 5 commits into
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torontoisolated
May 24, 2026
Merged
Toronto + not hyperconnected + finite => has an isolated point#1786
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mainfrom 
torontoisolated
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@felixpernegger
@felixperneggerfelixpernegger
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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
prabau
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
prabau
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
felixpernegger
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
felixpernegger
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
prabau
reviewed
May 24, 2026
View reviewed changes
Comment threadtheorems/T000898.md
Outdated

@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
into
mainMay 24, 2026

1 check passed

@prabau
prabau
deleted the

torontoisolated

branch
May 24, 2026 06:15
felixpernegger added a commit
that referenced
this pull request
May 24, 2026
@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
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, '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" + '

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Toronto + not hyperconnected + finite => has an isolated point - #1786
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torontoisolated
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Toronto + not hyperconnected + finite => has an isolated point#1786
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@felixpernegger
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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
prabau
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
prabau
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
felixpernegger
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
felixpernegger
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
prabau
reviewed
May 24, 2026
View reviewed changes
Comment threadtheorems/T000898.md
Outdated

@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
into
mainMay 24, 2026

1 check passed

@prabau
prabau
deleted the

torontoisolated

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May 24, 2026 06:15
felixpernegger added a commit
that referenced
this pull request
May 24, 2026
@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
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, '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");
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})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Toronto + not hyperconnected + finite => has an isolated point - #1786
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Merged
Toronto + not hyperconnected + finite => has an isolated point#1786
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torontoisolated
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@felixpernegger
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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
prabau
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
prabau
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
felixpernegger
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
felixpernegger
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
prabau
reviewed
May 24, 2026
View reviewed changes
Comment threadtheorems/T000898.md
Outdated

@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
into
mainMay 24, 2026

1 check passed

@prabau
prabau
deleted the

torontoisolated

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May 24, 2026 06:15
felixpernegger added a commit
that referenced
this pull request
May 24, 2026
@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
2447963
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, '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 {
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Toronto + not hyperconnected + finite => has an isolated point - #1786
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Toronto + not hyperconnected + finite => has an isolated point#1786
prabau merged 5 commits into
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torontoisolated
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@felixpernegger
@felixperneggerfelixpernegger
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May 23, 2026

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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
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May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
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May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
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May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
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Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
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Toronto + not hyperconnected + finite => has an isolated point (#1786)
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Skip to content
Toronto + not hyperconnected + finite => has an isolated point - #1786
Merged
prabau merged 5 commits into
mainfrom 
torontoisolated
May 24, 2026
Merged
Toronto + not hyperconnected + finite => has an isolated point#1786
prabau merged 5 commits into
mainfrom 
torontoisolated
Conversation
@felixpernegger
@felixperneggerfelixpernegger
commented
May 23, 2026

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We discussed in #1773, the author said he would PR this, but didnt do so (and didnt react to my follow up). Since this theorem is really powerful (over 50 traits!), I PR it myself.
@prabau
prabau
commented
May 24, 2026

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The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
@prabau
prabau
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
@felixpernegger
felixpernegger
commented
May 24, 2026

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I am curious. The only infinite Toronto spaces that are not hyperconnected currently in pi-base are discrete. Are there such spaces that are not discrete?
Under GCH infinite T2 spaces are Toronto and we dont really have many spaces which are not hyperconnected and not t2, see https://topology.pi-base.org/spaces?q=%7EHyperconnected+%2B+%7Et2

So this might just be a coincidence
@felixpernegger
felixpernegger
commented
May 24, 2026

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CollaboratorAuthor
The proof looks good.
Comparing with the list of theorems for Toronto (P219), would it be possible to move P219 to be the first hypothesis of the theorem?
I found this to be the most natural way to state it, but if you want me to change it, sure just say
prabau
prabau
reviewed
May 24, 2026
View reviewed changes
Comment threadtheorems/T000898.md
Outdated

@felixpernegger@prabau
Update theorems/T000898.md
ecdd0b3
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
@prabau
prabau
merged commit 7796ddc
into
mainMay 24, 2026

1 check passed

@prabau
prabau
deleted the

torontoisolated

branch
May 24, 2026 06:15
felixpernegger added a commit
that referenced
this pull request
May 24, 2026
@felixpernegger
Toronto + not hyperconnected + finite => has an isolated point (#1786)
2447963
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