Poincare Project: More on Separation Axioms

We previously defined types of topological spaces called T_0, T_1 and T_2 spaces. I've tried below to capture the distinction between them informally with a diagram.

Recall that a topological space is:

• T_0 iff for every pair of points x, y there is an open set U such that either x is in U but y is not or y is in U but x is not.
• T_1 iff for every pair of points x, y there are open sets U and V such that x is in U but y is not and y is in V but x is not.
• T_2 iff for every pair of points x, y there are open sets U and V such that x is in U, y is in V and U and V are disjoint.

UPDATE: next post