Show $\tau=\tau^$ if $\tau^\subset \tau$

Question

Let $(X,\tau)$ be compact and $(X,\tau^*)$ be a Hausdorff space. How can we show that $\tau=\tau^*$ if $\tau^*\subset \tau$?

Also, see Lemma 1 here: https://terrytao.wordpress.com/2009/02/09/245b-notes-10-compactness-in-topological-spaces/ — kahen, Mar 30 '13 at 15:52

score 0 · Answer 1 · answered Mar 30 '13 at 15:52

0

Hint: Consider the identity map $Id : (X,\tau) \rightarrow (X,\tau^*)$. Since $\tau^* \subset \tau$, this map is continuous. Also, a continuous map from a compact to a Hausdorff space is a closed.

answered Mar 30 '13 at 15:52

Ludolila

3,034

Show $\tau=\tau^*$ if $\tau^*\subset \tau$

1 Answers1

Show $\tau=\tau^$ if $\tau^\subset \tau$