Toronto spaces - roadmap

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)
+---
+
+Any 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/T000905.md

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+---
+uid: T000905
+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 $X$.

theorems/T000906.md

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+---
+uid: T000906
+if:
+  P000052: true
+then:
+  P000219: true
+---
+
+Let $Y\subseteq X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000907.md

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+---
+uid: T000907
+if:
+  P000222: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000908.md

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+---
+uid: T000908
+if:
+  P000129: true
+then:
+  P000219: true
+---
+
+Let $Y\subset X$ with $|Y|=|X|$. Then any bijection $Y \to X$ is a homeomorphism.

theorems/T000909.md

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+---
+uid: T000909
+if:
+  and:
+  - P000219: true
+  - P000204: true
+then:
+  P000078: true
+---
+
+Let $p$ be a cut point of $X$. If $X$ were infinite, $|X\setminus \{p\}|=|X|$, but they cannot be homeomorphic as $X$ is {P36} but $X \setminus \{p\}$ is not.
+

theorems/T000910.md

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+---
+uid: T000910
+if:
+  and:
+  - P000219: true
+  - P000196: false
+  - P000129: false
+  - P000078: false
+then:
+  P000002: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+Follows from Theorem 6.1 in {{zb:1286.54032}}.

theorems/T000911.md

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+---
+uid: T000911
+if:
+  and:
+  - P000219: true
+  - P000181: true
+  - P000129: false
+  - P000052: false
+  - P000222: false
+then:
+  P000196: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+Follows from Corollary 6.2 in {{zb:1286.54032}}.

theorems/T000912.md

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+---
+uid: T000912
+if:
+  and:
+  - P000219: true
+  - P000003: true
+then:
+  P000167: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+Immediate consequence of Theorem 3.1 in {{zb:1286.54032}}.

theorems/T000913.md

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+---
+uid: T000913
+if:
+  and:
+  - P000219: true
+  - P000003: true
+then:
+  P000093: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+See the proof of Corollary 3.4 in {{zb:1286.54032}}.

theorems/T000914.md

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+---
+uid: T000914
+if:
+  and:
+  - P000219: true
+  - P000136: false
+  - P000003: true
+then:
+  P000052: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+See Corollary 3.5 in {{zb:1286.54032}}.

theorems/T000918.md

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+---
+uid: T000918
+if:
+  and:
+  - P000219: true
+  - P000180: false
+  - P000003: true
+then:
+  P000052: true
+refs:
+  - zb: "1286.54032"
+    name: The Toronto Problem (W. R. Brian)
+---
+
+See Proposition 2.3 in {{zb:1286.54032}}.