Semiregular space - Revision history

Vipul at 17:01, 28 January 2012

2012-01-28T17:01:35Z

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	==Definition==		==Definition==

	A '''semiregular space''' is a [[T1 space]] ~~in which~~ the [[regular open subset]]s (these are subsets that equal the [[interior]] of their [[closure]]) form a [[basis]] for the space.		A '''semiregular space''' is a [[topological space]] <math>X</math>satisfying the following equivalent conditions:

			# The [[regular open subset]]s (these are subsets that equal the [[interior]] of their [[closure]]) form a [[basis]] for the space.
			# For any <math>x \in X</math> and any open subset <math>V \subseteq X</math> containing <math>x</math>, there exists a [[regular open subset]] <math>U</math> of <math>X</math> containing <math>x</math> and contained in <math>V</math>.

	Note that ''regular open'' is not the same as being open and regular in the [[subspace topology]]. For the notion defined using that, see [[locally regular space]].		Note that ''regular open'' is not the same as being open and regular in the [[subspace topology]]. For the notion defined using that, see [[locally regular space]].

Vipul: Created page with "{{topospace property}} ==Definition== A '''semiregular space''' is a T1 space in which the regular open subsets (these are subsets that equal the interior of the..."

2012-01-27T03:47:53Z

Created page with "{{topospace property}} ==Definition== A '''semiregular space''' is a T1 space in which the regular open subsets (these are subsets that equal the interior of the..."

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{{topospace property}}

==Definition==

A '''semiregular space''' is a [[T1 space]] in which the [[regular open subset]]s (these are subsets that equal the [[interior]] of their [[closure]]) form a [[basis]] for the space.

Note that ''regular open'' is not the same as being open and regular in the [[subspace topology]]. For the notion defined using that, see [[locally regular space]].

==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::regular space]] || || [[regular implies semiregular]] || [[semiregular not implies regular]] ||
|}