The equivalent definitions of ‘topology’…

From Armstrong:
On page 13, we have definition 1.3:
Given a set and for each point in a nonempty collection of subsets of , called neighborhoods of . These neighborhoods are required to satisfy four axioms:

lies in each of its neighborhoods.
The intersection of two neighborhoods of is also a neighborhood [...]

Every Topology is a Category!

This is just cool!
Okay, so we’ve defined a topology on a set . But we also love categories, so we want to see this in terms of categories. And, indeed, every topology is a category!

The Lyceum Mathematikoi Test Post

This is a test post.  This post is only a test.  Do not pass go, do not collect two-hundred dollars mod n.