Vipul at 04:34, 30 January 2014

2014-01-30T04:34:15Z

← Older revision		Revision as of 04:34, 30 January 2014
Line 2:		Line 2:
	property = resolvable space\|		property = resolvable space\|
	metaproperty = open subspace-closed property of topological spaces}}		metaproperty = open subspace-closed property of topological spaces}}
			[[Difficulty level::1\| ]]
	==Statement==		==Statement==

Vipul: Created page with "{{topospace metaproperty satisfaction| property = resolvable space| metaproperty = open subspace-closed property of topological spaces}} ==Statement== Suppose $X$ ..."

2012-01-26T20:10:57Z

Created page with "{{topospace metaproperty satisfaction| property = resolvable space| metaproperty = open subspace-closed property of topological spaces}} ==Statement== Suppose <math>X</math>..."

New page

{{topospace metaproperty satisfaction|
property = resolvable space|
metaproperty = open subspace-closed property of topological spaces}}

==Statement==

Suppose <math>X</math> is a [[resolvable space]] and <math>U</math> is an [[open subset]] of <math>X</math>. Then, <math>U</math> is a resolvable space with the subspace topology.

==Facts used==

# [[uses::Intersection of dense subset with open subset is dense in the open subset]]

==Proof==

'''Given''': A topological space <math>X</matH> with disjoint dense subsets <math>C</math> and <math>D</math>. An open subset <math>U</math> of <math>X</math>.

'''To prove''': <math>U</math> has two disjoint dense subsets.

'''Proof''':

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || <math>C \cap U</math> and <math>D \cap U</math> are both dense in <math>U</math>. || Fact (1) || <math>C,D</math> dense in <math>X</math><br><math>U</math> open in <math>X</math> || || given-fact direct
|-
| 2 || <math>C \cap U</math> and <math>D \cap U</math> are disjoint. || || <math>C,D</math> are disjoint || || basic set theory!
|-
| 3 || <math>C \cap U</math> and <math>D \cap U</math> are the desired disjoint dense subsets in <math>U</math> || || || Steps (1), (2) || Step-combination direct
|}

Resolvability is open subspace-closed - Revision history

Vipul at 04:34, 30 January 2014

Vipul: Created page with "{{topospace metaproperty satisfaction| property = resolvable space| metaproperty = open subspace-closed property of topological spaces}} ==Statement== Suppose X..."

Vipul: Created page with "{{topospace metaproperty satisfaction| property = resolvable space| metaproperty = open subspace-closed property of topological spaces}} ==Statement== Suppose $X$ ..."