Regular Hausdorff space - Revision history

Vipul: /* Definition */

2012-01-28T02:04:07Z

Definition

← Older revision		Revision as of 02:04, 28 January 2012
Line 5:		Line 5:
	# It is both a [[regular space]] and a [[defining ingredient::Hausdorff space]].		# It is both a [[regular space]] and a [[defining ingredient::Hausdorff space]].
	# It is both a [[regular space]] and a [[defining ingredient::T1 space]].		# It is both a [[regular space]] and a [[defining ingredient::T1 space]].
			# It is both a [[regular space]] and a [[defining ingredient::Kolmogorov space]] (i.e., a <math>T_0</math> space).

	Note that outside of point-set topology, and in many elementary treatments, the term ''regular space'' is used to stand for regular Hausdorff space.		Note that outside of point-set topology, and in many elementary treatments, the term ''regular space'' is used to stand for regular Hausdorff space.

Vipul: /* Weaker properties */

2012-01-27T21:51:59Z

Weaker properties

← Older revision		Revision as of 21:51, 27 January 2012
Line 50:		Line 50:
	\|-		\|-
	\| [[Stronger than::T1 space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|T1 space\|regular Hausdorff space}}		\| [[Stronger than::T1 space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|T1 space\|regular Hausdorff space}}
			\|-
			\| [[Stronger than::Kolmogorov space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Kolmogorov space\|regular Hausdorff space}}
	\|}		\|}

Vipul: /* Relation with other properties */

2012-01-27T21:51:16Z

Relation with other properties

@@ Line 38: / Line 38: @@
 |-
 | [[Weaker than::paracompact Hausdorff space]] || || || || {{intermediate notions short|regular Hausdorff space|paracompact Hausdorff space}}
 |}

Vipul at 21:49, 27 January 2012

2012-01-27T21:49:00Z

← Older revision		Revision as of 21:49, 27 January 2012
Line 11:		Line 11:

	{{T family\|T3}}		{{T family\|T3}}

			==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::metrizable space]] \|\| underlying topological space of a [[metric space]] \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|metrizable space}}
			\|-
			\| [[Weaker than::CW-space]] \|\| topological space arising as the underlying space of a [[CW-complex]] \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|CW-space}}
			\|-
			\| [[Weaker than::perfectly normal Hausdorff space]] \|\| it is [[normal space\|normal]] and every [[closed subset]] is a [[G-delta subset]] \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|perfectly normal Hausdorff space}}
			\|-
			\| [[Weaker than::hereditarily normal Hausdorff space]] \|\| every subset is normal in the subspace topology \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|hereditarily normal Hausdorff space}}
			\|-
			\| [[Weaker than::monotonically normal Hausdorff space]] \|\| (follow link for definition) \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|monotonically normal Hausdorff space}}
			\|-
			\| [[Weaker than::normal Hausdorff space]] \|\| T1 and disjoint closed subsets can be separated by disjoint open subsets \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|normal Hausdorff space}}
			\|-
			\| [[Weaker than::Tychonoff space]] \|\| T1 and point and closed subset not containing it can be separated by continuous function \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|Tychonoff space}}
			\|-
			\| [[Weaker than::compact Hausdorff space]] \|\| [[compact space\|compact]] and [[Hausdorff space\|Hausdorff]] \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|compact Hausdorff space}}
			\|-
			\| [[Weaker than::locally compact Hausdorff space]] \|\| [[locally compact space\|locally compact]] and [[Hausdorff space\|Hausdorff]] \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|locally compact Hausdorff space}}
			\|-
			\| [[Weaker than::paracompact Hausdorff space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|regular Hausdorff space\|paracompact Hausdorff space}}
			\|}

Vipul: Created page with "==Definition== A topological space is termed a '''regular Hausdorff space''' or a $T_3$ space if it satisfies the following equivalent conditions: # It is bot..."

2012-01-27T17:58:10Z

Created page with "==Definition== A topological space is termed a '''regular Hausdorff space''' or a <math>T_3</math> space if it satisfies the following equivalent conditions: # It is bot..."

New page

==Definition==

A [[topological space]] is termed a '''regular Hausdorff space''' or a <math>T_3</math> space if it satisfies the following equivalent conditions:

# It is both a [[regular space]] and a [[defining ingredient::Hausdorff space]].
# It is both a [[regular space]] and a [[defining ingredient::T1 space]].

Note that outside of point-set topology, and in many elementary treatments, the term ''regular space'' is used to stand for regular Hausdorff space.

{{topospace property}}

{{T family|T3}}

Regular Hausdorff space - Revision history

Vipul: /* Definition */

Vipul: /* Weaker properties */

Vipul: /* Relation with other properties */

Vipul at 21:49, 27 January 2012

Vipul: Created page with "==Definition== A topological space is termed a '''regular Hausdorff space''' or a T_3 space if it satisfies the following equivalent conditions: # It is bot..."

Vipul: Created page with "==Definition== A topological space is termed a '''regular Hausdorff space''' or a $T_3$ space if it satisfies the following equivalent conditions: # It is bot..."