00:01
Hi in this question we have been asked to show that every compact space is locally compact.
00:08
So we have to consider a topological space.
00:11
So let us say that consider the topological space and that is x comma tau.
00:25
And therefore we will say that x comma tau this is compact.
00:30
Then we have to prove that x comma tau this is locally compact we will show this by using the hosdrop space so now we'll say that if x is hosdrop and compact so this implies that x will also be normal so in particular this is equivalent to each point having a neighborhood basis and consisting of a close neighborhood as it is compact.
01:39
So this implies so that we will be getting that this is a close neighborhood.
01:57
This implies it is compact.
02:01
So now we say if x belongs to capital x, this has a neighborhood basis consisting of the compact space, compact sets.
02:11
So this consists the compact set.
02:18
Then this implies that the set a topology x, it is also locally compact.
02:29
Now earlier we have considered that x was hosdrop...
