Vipul: /* Definitions used */

2012-01-28T17:17:53Z

Definitions used

← Older revision		Revision as of 17:17, 28 January 2012
Line 12:		Line 12:
	! Term !! Definition used		! Term !! Definition used
	\|-		\|-
	\| [[regular space]] \|\| A [[topological space]] <math>X</math> is termed regular if the following holds: ~~given any point and a [[closed subset]] not containing it, there are disjoint open subsets containing them. \|\|~~ given any point <math>x \in X</math> and open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U \subseteq X</math> such that <math>x \in U</math> and the [[closure]] <math>\overline{U}</math> is contained in <math>V</math>.		\| [[regular space]] \|\| A [[topological space]] <math>X</math> is termed regular if the following holds: given any point <math>x \in X</math> and open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U \subseteq X</math> such that <math>x \in U</math> and the [[closure]] <math>\overline{U}</math> is contained in <math>V</math>.
	\|-		\|-
	\| [[semiregular space]] \|\| A [[topological space]] <math>X</math> is termed regular if the following holds: given 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'' does not mean an open subset that is regular in the subspace topology. Rather, it means a subset that is the [[interior]] of its [[closure]].		\| [[semiregular space]] \|\| A [[topological space]] <math>X</math> is termed regular if the following holds: given 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'' does not mean an open subset that is regular in the subspace topology. Rather, it means a subset that is the [[interior]] of its [[closure]].

Vipul: Created page with "{{topospace property implication| stronger = regular space| weaker = semiregular space}} ==Statement== Any regular space is a semiregular space. ==Definitions used=..."

2012-01-28T17:17:26Z

Created page with "{{topospace property implication| stronger = regular space| weaker = semiregular space}} ==Statement== Any regular space is a semiregular space. ==Definitions used=..."

New page

{{topospace property implication|
stronger = regular space|
weaker = semiregular space}}

==Statement==

Any [[regular space]] is a [[semiregular space]].

==Definitions used==

{| class="sortable" border="1"
! Term !! Definition used
|-
| [[regular space]] || A [[topological space]] <math>X</math> is termed regular if the following holds: given any point and a [[closed subset]] not containing it, there are disjoint open subsets containing them. || given any point <math>x \in X</math> and open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U \subseteq X</math> such that <math>x \in U</math> and the [[closure]] <math>\overline{U}</math> is contained in <math>V</math>.
|-
| [[semiregular space]] || A [[topological space]] <math>X</math> is termed regular if the following holds: given 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'' does not mean an open subset that is regular in the subspace topology. Rather, it means a subset that is the [[interior]] of its [[closure]].
|}

==Proof==

'''Given''': A regular space <math>X</math>. A point <math>x \in X</math> and any open subset <math>V \subseteq X</math> containing <math>x</math>,

'''To prove''': There exists a regular open subset <math>U</math> of <math>X</math> containing <math>x</math> and contained in <math>V</math>.

'''Proof''':

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Given data used !! Previous steps used !! Explanation
|-
| 1 || There exists an open subset <math>U_0</math> containing <math>x</math> such that <math>\overline{U_0}</math> is contained in <math>V</math>. || <math>X</math> is regular, <math>x \in V</math>, <math>V</math> open in <math>X</math> || || directly from regularity.
|-
| 2 || Let <math>U</math> be the interior of the closure <math>\overline{U_0}</math>. || || Step (1) ||
|-
| 3 || <math>U</math> contains <math>U_0</math> and hence contains <math>x</math>. || || Steps (1), (2) || <math>U</math> is the largest open subset of <math>\overline{U_0}</math>, hence contains the open subset <math>U_0</math>.
|-
| 4 || <math>U</math> is a regular open subset. || || Step (2) || step-direct
|-
| 5 || <math>U</math> is contained in <math>V</math>. || || Steps (1), (2) || By Step (2), <math>U \subseteq \overline{U_0}</math>. By Step (1), <math>\overline{U_0} \subseteq V</math>. Combining, <math>U \subseteq V</math>.
|-
| 6 || <math>U</math> is the desired open subset. || || Steps (3), (4), (5) ||
|}

Regular implies semiregular - Revision history

Vipul: /* Definitions used */

Vipul: Created page with "{{topospace property implication| stronger = regular space| weaker = semiregular space}} ==Statement== Any regular space is a semiregular space. ==Definitions used=..."