**Contents**

**Overview**

**ProvenMath** > **Topology** >
Topological Space

This page defines the following concepts: topological space, open set, closed set, interior, closure, and states the most basic theorems about these concepts with complete elementary proofs. For example, it contains basic facts like clo(A u B) = clo(A) u clo(B), or int(A) = A if and only if A is open, etc.

**ProvenMath** > **Topology** >
Subspace of Topological Space

This page defines the concept of a topological subspace and provides all the basic theorems for dealing with subspaces. Interestingly, it is quite easy to convince oneself that the definition of the relative topology satisfies the axioms for a topological space. However, the easy straightforward proof uses the Axiom of Choice in a way that is almost transparent to an unexperienced eye. This page proves this simple well-known fact in a slightly long and meticulous way in order to avoid the use of the Axiom of Choice.

**ProvenMath** > **Topology** >
Separation Axioms

The page contains a very precise formulation
of the following topological concepts:
T_0 space, T_1 space, T_2 space, T_3 space, T_4 space, T_5 space,
Hausdorff space, regular space, and normal space.
It includes basic theorems showing appropriate implications between these separation properties.
For more detailed information about separation conditions see the recommended book
*Counterexamples in Topology*
by Lynn Arthur Steen and J. Arthur Seebach, Jr. - which lists all separation conditions
(including T_2.5, T_3.5, semiregular, completely Hausdorff, perfectly normal, etc.)
and shows all implications that hold between them and gives counterexamples for the implications
that do not hold in general.