A description/proof of that for topological space, subset of compact subset is not necessarily compact
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 Euclidean topology.
Target Context
- The reader will have a description and a proof of the proposition that for a topological space, a subset of a compact subset is not necessarily compact.
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, \(T\), and a compact subset, \(S \subseteq T\), a subset, \(S_1 \subseteq S\), is not necessarily compact on \(T\).
2: Proof
A counterexample suffices. Take \(T = \mathbb{R}^2\) with the Euclidean topology, \(S = \overline{B_{p-\epsilon}}\), and \(S_1 = B_{p-\epsilon}\) where \(B_{p-\epsilon}\) is the \(\epsilon\)-radius open ball centered at \(p\) and the over line denotes the closure.
3: Note
The word, "compact", may sound like about being small, but a smaller subset of a compact subset is not necessarily compact.
