A description/proof of that subset of 1st category subset is of 1st category
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of 1st category subset.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, any subset of any 1st category subset is of the 1st category.
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 1st category subset, \(S \subseteq T\), any subset, \(S' \subseteq S\), is of the 1st category.
2: Proof
\(S = \cup_{i \in I} S_i\) where \(I\) is any countable indices set and \(S_i\) is any nowhere dense subset. For \(S'' = S \setminus S'\), \(S' = \cup_{i \in I} (S_i \setminus S'')\), while \(S_i \setminus S''\) is nowhere dense.
