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felixpernegger merged 5 commits into from
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Toronto spaces definition + easy properties #1564

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Toronto spaces itroduction + easy properties
felixpernegger Dec 28, 2025
b620f84
resolve merge conflict
felixpernegger Dec 28, 2025
84ec027
any -> every typo
felixpernegger Dec 28, 2025
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apply suggestions
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13 changes: 13 additions & 0 deletions properties/P000219.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
- wikipedia: Toronto_space
name: Toronto space on Wikipedia
- zb: "1286.54032"
name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.
9 changes: 9 additions & 0 deletions theorems/T000814.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
P000129: true
then:
P000219: true
---

Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
11 changes: 11 additions & 0 deletions theorems/T000815.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
and:
- P000219: true
- P000078: false
then:
P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.
9 changes: 9 additions & 0 deletions theorems/T000816.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
P000222: true
then:
P000219: true
---

Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
9 changes: 9 additions & 0 deletions theorems/T000817.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
P000052: true
then:
P000219: true
---

Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.
9 changes: 9 additions & 0 deletions theorems/T000818.md
View file
Open in desktop
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
P000078: true
then:
P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.
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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces itroduction + easy properties


felixpernegger Dec 28, 2025
b620f84

resolve merge conflict


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84ec027

any -> every typo


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

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces definition + easy properties#1564

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Toronto spaces itroduction + easy properties


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b620f84

resolve merge conflict


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84ec027

any -> every typo


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c6eb101

apply suggestions


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

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

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// Highlight search terms from Google/DuckDuckGo/Bing referrer
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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces definition + easy properties#1564

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felixpernegger
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Dec 29, 2025
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Toronto spaces definition + easy properties
#1564
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e9b4937

Toronto spaces itroduction + easy properties


felixpernegger Dec 28, 2025
b620f84

resolve merge conflict


felixpernegger Dec 28, 2025
84ec027

any -> every typo


felixpernegger Dec 28, 2025
c6eb101

apply suggestions


felixpernegger Dec 29, 2025
e6f5d6e

apply suggestion


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

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces definition + easy properties#1564

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 Merged

felixpernegger
merged 5 commits into

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Dec 29, 2025
 Merged


Toronto spaces definition + easy properties
#1564
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e9b4937

Toronto spaces itroduction + easy properties


felixpernegger Dec 28, 2025
b620f84

resolve merge conflict


felixpernegger Dec 28, 2025
84ec027

any -> every typo


felixpernegger Dec 28, 2025
c6eb101

apply suggestions


felixpernegger Dec 29, 2025
e6f5d6e

apply suggestion


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

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces itroduction + easy properties


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resolve merge conflict


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any -> every typo


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properties/P000219.md

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Original file line numberDiff line numberDiff line change
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---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces definition + easy properties#1564

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#1564
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Toronto spaces itroduction + easy properties


felixpernegger Dec 28, 2025
b620f84

resolve merge conflict


felixpernegger Dec 28, 2025
84ec027

any -> every typo


felixpernegger Dec 28, 2025
c6eb101

apply suggestions


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

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.

, 'i');
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Toronto spaces definition + easy properties by felixpernegger · Pull Request #1564 · pi-base/data · GitHub
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Toronto spaces definition + easy properties#1564

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 Merged

felixpernegger
merged 5 commits into

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Dec 29, 2025
 Merged


Toronto spaces definition + easy properties
#1564
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e9b4937

Toronto spaces itroduction + easy properties


felixpernegger Dec 28, 2025
b620f84

resolve merge conflict


felixpernegger Dec 28, 2025
84ec027

any -> every typo


felixpernegger Dec 28, 2025
c6eb101

apply suggestions


felixpernegger Dec 29, 2025
e6f5d6e

apply suggestion


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

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,13 @@
---
uid: P000219
name: Toronto
refs:
 - wikipedia: Toronto_space
 name: Toronto space on Wikipedia
 - zb: "1286.54032"
 name: The Toronto Problem (W. R. Brian)
---

Every subspace $Y \subseteq X$ with $|Y|=|X|$ is homeomorphic to $X$.

In {{zb:1286.54032}} it is shown that under GCH, every {P3} Toronto space is {P52}.

9 changes: 9 additions & 0 deletions
theorems/T000814.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000814
if:
 P000129: true
then:
 P000219: true
---

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

11 changes: 11 additions & 0 deletions
theorems/T000815.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,11 @@
---
uid: T000815
if:
 and:
 - P000219: true
 - P000078: false
then:
 P000204: false
---

Assume $X$ has a cut point $p$. Then $|X\setminus \{p\}|=|X|$, but the two spaces cannot be homeomorphic as $X$ is {P36} and $X \setminus \{p\}$ is not.

9 changes: 9 additions & 0 deletions
theorems/T000816.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000816
if:
 P000222: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000817.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000817
if:
 P000052: true
then:
 P000219: true
---

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

9 changes: 9 additions & 0 deletions
theorems/T000818.md

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,9 @@
---
uid: T000818
if:
 P000078: true
then:
 P000219: true
---

For a finite space $X$, the only subspace with the same cardinality is $X$ itself, which is trivally homeomorphic to $X$.