A description/proof of that for topological space, intersection of compact subset and subspace is not necessarily compact on subspace
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact subset of topological space.
- The reader knows a definition of subspace topology.
- The reader knows a definition of Euclidean topology.
Target Context
- The reader will have a description and a proof of the proposition that for a topological space, the intersection of a compact subset and a subspace is not necessarily compact on the subspace.
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 a topological space, \(T\), a compact subset, \(S \subseteq T\), and a subspace, \(T_1 \subseteq T\), the intersection, \(S \cap T_1\), is not necessarily compact on \(T_1\).
2: Proof
A counterexample suffices. Take \(T = \mathbb{R}^2\) with the Euclidean topology, \(S = \overline{B_{p-\epsilon}}\), and \(T_1 = B_{p-\epsilon}\) where \(B_{p-\epsilon}\) is the \(\epsilon\)-radius open ball centered at \(p\) and the over line denotes the closure. \(S \cap T_1 = B_{p-\epsilon}\) is not compact on \(B_{p-\epsilon}\).
3: Note
If \(S \subseteq T_1\), \(S\) is necessarily compact on \(T_1\), by the proposition that for any topological space, any subspace subset that is compact on base space is compact on the subspace. Let us note the difference.
