Vipul: /* Definitions used */

2012-01-28T16:42:30Z

Definitions used

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	\| [[preregular space]] \|\| A topological space <math>X</math> is termed preregular if the following holds: given any [[topologically distinguishable points]] <math>x,y \in X</math>, there exist disjoint open subsets <math>U \ni x, V \ni y</math>.		\| [[preregular space]] \|\| A topological space <math>X</math> is termed preregular if the following holds: given any [[topologically distinguishable points]] <math>x,y \in X</math>, there exist disjoint open subsets <math>U \ni x, V \ni y</math>.
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	==Proof==		==Proof==

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

2012-01-28T16:42:04Z

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

New page

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

==Statement==

Any [[regular space]] is a [[preregular 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 <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>x \in U, A \subseteq V</math>, and <math>U \cap V = \varnothing</math>.
|-
| [[preregular space]] || A topological space <math>X</math> is termed preregular if the following holds: given any [[topologically distinguishable points]] <math>x,y \in X</math>, there exist disjoint open subsets <math>U \ni x, V \ni y</math>.

==Proof==

'''Given''': A topological space <math>X</math> that is regular. Topologically distinguishable points <math>x,y \in X</math>.

'''To prove''': There exist disjoint open subsets <math>U \ni x, V \ni y</math>.

'''Proof''': Because <math>x,y</matH> are topologically distinguishable, it is true that ''either'' <math>x \notin \overline{ \{ y \} }</math> or <math>y \notin \overline{ \{ x \} }</math> (both may also be true). it suffices to consider only the first case, because the result we want to prove is symmetric in <math>x</math> and <math>y</math>.

In the first case, let <math>A = \overline{ \{ y \} }</math>. Now use the definition of regularity on <math>x</math> and <math>A</math> to obtain the open subsets <math>U</math> and <math>V</math> satisfying the desired conditions.

Regular implies preregular - Revision history

Vipul: /* Definitions used */

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