How do I reduce a countable subcover to a finite subcover

[Cratylus]

This is general question relating to compactness. I am not to sure on the method
So for sake of argument,let X be a 2nd countable set. By definition, I have a countable basis .
Let $B_n$ ={$B_1,…, B_n$} be that basis. Then for each n$\in$ Z

How do I reduce a countable subcover to a finite subcover ?

 

[pka]

This is general question relating to compactness. I am not to sure on the method
So for sake of argument,let X be a 2nd countable set. By definition, I have a countable basis .
Let $B_n$ ={$B_1,…, B_n$} be that basis. Then for each n$\in$ Z

How do I reduce a countable subcover to a finite subcover ?

You are miss-using the the definition of first & second countable!
See General Topology by Helen F. Cullen, page 33.
A topological space [imath](\mathcal{S},\mathfrak{T})[/imath] is said to be a first countable space iff each point [imath]x\in\mathcal{S}[/imath] has a countable local base.
[imath](\mathcal{S},\mathfrak{T})[/imath] is second countable iff there exists a countable base for the topology [imath]\mathfrak{T}[/imath].
So first and second countable spaces do not necessarily have the finite subcover property.
Please read your source material again. It seems yo may be confusing this with compactness.

 

[Cratylus]

You are miss-using the the definition of first & second countable!
See General Topology by Helen F. Cullen, page 33.
A topological space [imath](\mathcal{S},\mathfrak{T})[/imath] is said to be a first countable space iff each point [imath]x\in\mathcal{S}[/imath] has a countable local base.
[imath](\mathcal{S},\mathfrak{T})[/imath] is second countable iff there exists a countable base for the topology [imath]\mathfrak{T}[/imath].
So first and second countable spaces do not necessarily have the finite subcover property.
Please read your source material again. It seems yo may be confusing this with compactness.

Wow I didn’t finish writing it and got an answer.
I know the definitions. The definition of 2nd countable is in the section on Compactness in my text .So I assumed it dealt with compactness
My text is first course in topology by Robert conover ,1975