Does compact normal space implies metric space?

Question

I know compact Hausdorff space is normal, and metric space is normal.

What about compact normal space? Is it a metric space?

My naive guess is that normality is nothing to do with metric and compactness(finiteness) is also nothing to do with metric so it seems it is not a metric space. But how one can prove or disprove this more than my idea?

Answer 1

A beautiful example to the contrary is the Double Arrow Space, one of the classics in topology.

It's compact, perfectly normal ($T_6$), hereditarily Lindelöf, hereditarily separable, first countable, but not metrisable as it does not have a countable base.