Tychonoff space - Revision history

Vipul at 20:00, 29 January 2014

2014-01-29T20:00:18Z

← Older revision		Revision as of 20:00, 29 January 2014
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	{{T family\|T3.5}}		{{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}}
			\|}

Vipul: /* Definition */

2012-01-27T18:02:25Z

Definition

← Older revision		Revision as of 18:02, 27 January 2012
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	\| 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>.		\| 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>.		\| 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>.
	\|-		\|-

Vipul: Created page with "==Definition== A topological space is termed a '''Tychonoff space''' if it satisfies the following equivalent conditions: {| class="sortable" border="1" ! No. !! Shortha..."

2012-01-27T18:01:04Z

Created page with "==Definition== A topological space is termed a '''Tychonoff space''' if it satisfies the following equivalent conditions: {| class="sortable" border="1" ! No. !! Shortha..."

New page

==Definition==

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

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is termed Tychonoff if ... !! A topological space <math>X</math> is termed Tychonoff if ...
|-
| 1 || T1 and completely regular || it is both a [[T1 space]] and a [[completely regular space]] || points are closed in <math>X</math>, 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>.
|-
| 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>.
|-
| 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]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
|}

{{topospace property}}

{{T family|T3.5}}