TITLE: Non-Metrizable Topological Spaces
QUESTION [8 upvotes]: What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. Also, it would be cool and informative if you could list some basic topological properties that each of these spaces have. Thanks.

REPLY [3 votes]: $\pi$-Base lists ninety-five non-metrizable spaces. Rather than include them all in this post, I will simply link you to the search result.