A description/proof of that superset of residual subset is residual
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of residual subset.
- The reader admits the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, any superset of any residual subset is residual.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological space, \(T\), and any residual subset, \(S \subseteq T\), any superset, \(S'\), such that \(S \subseteq S'\), is residual.
2: Proof
\(S = T \setminus S_0\) where \(S_0\) is of the 1st category. \(S'' := S' \setminus S\) where \(S'' \subseteq T \setminus S = S_0\). \(S' = S \cup S'' = (T \setminus S_0) \cup S'' = T \setminus (S_0 \setminus S'')\). \(S_0 \setminus S''\) is of the 1st category, by the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.
