description/proof of that 2nd-countable topological space is compact iff it is countably compact
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of 2nd-countable topological space.
- The reader knows a definition of compact topological space.
- The reader knows a definition of countably compact topological space.
- The reader admits the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
Target Context
- The reader will have a description and a proof of the proposition that any 2nd-countable topological space is compact if and only if it is countably 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(T\): \(\in \{\text{ the 2nd-countable topological spaces }\}\)
//
Statements:
\(T \in \{\text{ the compact topological spaces }\}\)
\(\iff\)
\(T \in \{\text{ the countably compact topological spaces }\}\)
//
2: Proof
Whole Strategy: apply the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover; Step 1: suppose that \(T\) is compact; Step 2: see that \(T\) is countably compact; Step 3: suppose that \(T\) is countably compact; Step 4: see that \(T\) is compact.
Step 1:
Let us suppose that \(T\) is compact.
Step 2:
\(T\) is countably compact, because for any countable open cover of \(T\), it is an open cover of \(T\), so, there is a finite subcover, because \(T\) is compact.
Step 3:
Let us suppose that \(T\) is countably compact.
Step 4:
For any open cover of \(T\), there is a countable subcover, by the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
As \(T\) is countably compact, there is a finite subcover of the countable subcover.
But the finite subcover is a finite subcover of the original open cover.
So, \(T\) is compact.
