A description/proof of that open set intersects subset if it intersects closure of subset
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of closure of subset.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, any open set intersects any subset if it intersects the closure of the subset.
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 subset, \(S \subseteq T\), any open set, \(U\), intersects \(S\) if \(U\) intersects the closure of \(S\), \(\overline{S}\).
2: Proof
Suppose that \(U \cap \overline{S} \neq \emptyset\) and \(U \cap S = \emptyset\). \(S \subseteq (T \setminus U)\) and \(\overline{S} \cap (T \setminus U) \subset \overline{S}\). So, \(S \subseteq \overline{S} \cap (T \setminus U) \subset \overline{S}\), a contradiction, because \(\overline{S} \cap (T \setminus U)\) would be a closed set that contains \(S\), which (the closed set) would be smaller than the closure of \(S\). So, \(U \cap S \neq \emptyset\) if \(U \cap \overline{S} \neq \emptyset\).
