Q:

Topologically simple compact topological groups

In a lecture note, there is the following sentence without any reference or proofs:

Is there any reference or proofs for it?

A:

First of all, by a compact topological group one usually understands a topological group which, is a compact Hausdorff topological space. I think, Hausdorffness assumption here is due to Bourbaki who always assume  compact topological spaces to be Hausdorff (without Hausdorffness assumption they use the terminology quasicompact). With this in mind, you can find a proof of the fact that every topologically simple compact group is abstractly simple, for instance, in Theorem 9.90 of 
Karl H. Hofmann, Sidney A. Morris, 
The Structure of Compact Groups. 
A Primer for the Student – A Handbook for the Expert, 
de Gruyter Studies in Mathematics, Volume 25, 1998. 
Without Hausdorffness, the claim is clearly false since every group can be equipped with the trivial topology which makes it compact. Such a group will be topologically simple (since it contains only two closed subsets) but not necessarily algebraically simple. Consider, for instance, the additive group of integers. 
Edit. One can replace Hausdorff by T1 since a topological group is Hausdorff if and only if it is T1.

