Toronto spaces definition + easy properties (#1564)

Author: felixpernegger

properties/P000219.md

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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}.

theorems/T000814.md

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+---
+uid: T000814
+if:
+  P000129: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000815.md

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+---
+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.

theorems/T000816.md

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+---
+uid: T000816
+if:
+  P000222: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000817.md

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+---
+uid: T000817
+if:
+  P000052: true
+then:
+  P000219: true
+---
+
+Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000818.md

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+---
+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 $X$.