Tychonoff space: Difference between revisions

Latest revision as of 20:00, 29 January 2014

Definition

A topological space is termed a Tychonoff space if it satisfies the following equivalent conditions:

No.	Shorthand	A topological space is termed Tychonoff if ...	A topological space $X$ is termed Tychonoff if ...
1	T1 and completely regular	it is both a T1 space and a completely regular space	points are closed in $X$ , and given any point $x \in X$ and closed subset $A \subseteq X$ such that $x \notin A$ , there exists a continuous map $f : X \to [0, 1]$ such that $f (x) = 0$ and $f (a) = 1$ for all $a \in A$ .
2	Hausdorff and completely regular	it is both a Hausdorff space and a completely regular space	it is Hausdorff, and given any point $x \in X$ and closed subset $A \subseteq X$ such that $x \notin A$ , there exists a continuous map $f : X \to [0, 1]$ such that $f (x) = 0$ and $f (a) = 1$ for all $a \in A$ .
3	has a compactification	there is a compact Hausdorff space having a dense subspace (with the subspace topology) homeomorphic to it. (note: T1 assumption redundant in this case)	there is a compact Hausdorff space $C$ and a dense subspace $U$ of $X$ such that $X$ is homeomorphic to $U$ .
4	contained in compact Hausdorff	it is homeomorphic to a subspace (not necessarily dense) of a compact Hausdorff space (note: T1 assumption redundant in this case).	there is a compact Hausdorff space $C$ and a subspace $U$ of $X$ such that $X$ is homeomoephic to $U$ .

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T3.5

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
normal Hausdorff space	Hausdorff and a normal space: disjoint closed subsets can be separated via disjoint open subsets		\|FULL LIST, MORE INFO
compact Hausdorff space	Hausdorff and a compact space: every open cover has a finite subcover	via compact Hausdorff implies normal	Paracompact Hausdorff space\|FULL LIST, MORE INFO
underlying space of T0 topological group			\|FULL LIST, MORE INFO
metrizable space	underlying topological space of a metric space		Paracompact Hausdorff space\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
completely regular space				\|FULL LIST, MORE INFO
regular Hausdorff space				\|FULL LIST, MORE INFO
regular space				Regular Hausdorff space\|FULL LIST, MORE INFO
Hausdorff space				Functionally Hausdorff space, Regular Hausdorff space, Urysohn space\|FULL LIST, MORE INFO

@@ Line 10: / Line 10: @@
 | 2 || Hausdorff and completely regular || it is both a [[Hausdorff space]] and a [[completely regular space]] || it is Hausdorff, and given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.
 |-
-|
 | 3 || has a [[defining ingredient::compactification]] || there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism|homeomorphic]] to it. (note: T1 assumption redundant in this case) || there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomorphic to <math>U</math>.
 |-
@@ Line 19: / Line 18: @@
 {{T family|T3.5}}
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::normal Hausdorff space]] ||  Hausdorff and a [[normal space]]: disjoint closed subsets can be separated via disjoint open subsets || || || {{intermediate notions short|Tychonoff space|normal Hausdorff space}}
+|-
+| [[Weaker than::compact Hausdorff space]] || Hausdorff and a [[compact space]]: every open cover has a finite subcover || via [[compact Hausdorff implies normal]] || || {{intermediate notions short|Tychonoff space|compact Hausdorff space}}
+|-
+| [[Weaker than::underlying space of T0 topological group]] || || || || {{intermediate notions short|Tychonoff space|underlying space of T0 topological group}}
+|-
+| [[Weaker than::metrizable space]] || underlying topological space of a [[metric space]] || || || {{intermediate notions short|Tychonoff space|metrizable space}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::completely regular space]] || || || || {{intermediate notions short|Tychonoff space|completely regular space}}
+|-
+| [[Stronger than::regular Hausdorff space]] || || || || {{intermediate notions short|regular Hausdorff space|Tychonoff space}}
+|-
+| [[Stronger than::regular space]] || || || || {{intermediate notions short|regular space|Tychonoff space}}
+|-
+| [[Stronger than::Hausdorff space]] || || || || {{intermediate notions short|Hausdorff space|Tychonoff space}}
+|}