Resolvability is open subspace-closed
This article gives the statement, and possibly proof, of a topological space property (i.e., resolvable space) satisfying a topological space metaproperty (i.e., open subspace-closed property of topological spaces)
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Given: A topological space with disjoint dense subsets and . An open subset of .
To prove: has two disjoint dense subsets.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||and are both dense in .||Fact (1)|| dense in
|2||and are disjoint.||are disjoint||basic set theory!|
|3||and are the desired disjoint dense subsets in||Steps (1), (2)||Step-combination direct|