This question is from an assignment which I tried few months earlier but couldn't solve it. I got ill and couldn't ask back then.
If (X,T) is a topological space , the cone over X is the quotient space $(X\times I)$/ ~, where ~ is the equivalence relation on X$\times$ I defined by (a,1)~ (b,1) for all a,b $\in $X, for all a,b $\in $X and if t$\neq$1 ,(a,t) ~ (b,s) iff a=b and s=t. Prove that if the cone over a topological space (X,T) is locally compact, then the cone over X is compact.
To prove X$\times $ I is compact let there exists an open cover $ \bigcup_iO_i$ covering X$\times$ I /~ .
BuT i am at loss of ideas on how to prove that there exists a finite subcover.
Can you please help with that.