Following on from my post last week about metric spaces, I thought today I would describe what a topological space is.
Definition
A Topological Space (X, τ) consists of a set C and a set (the topology) τ of subsets of X such that:
 ∅, X ∈ U;
 If Ui ∈ τ for all i ∈ I, then ;

If U1, U2 ∈ τ, then U1 ∩ U2 ∈ τ.
The closure property (#3) is true for all finite intersections.
The elements of X are the points and the elements of τ are the open subsets of X.
Note that a subset Y of X in a topological space (X, τ) is called closed if X\Y is open, so we can describe a topology on a set X by specifying the closed sets in X which satisfy:
 ∅, X are closed;
 If Fi is closed for all i ∈ I, then so is ;

If F1, F2 are closed, then so is F1 ∪ F2.
Metric Topologies
A topology can be induced by a metric space (X,d); These are called metric topologies.
For example, the discrete metric on a set X gives rise to the discrete topology in which every subset in X is open, i.e. τ is the power set of X.
Example of NonMetric Topologies
 Let X be a set with at least two elements, and τ := {X,∅}. This is called the indiscrete topology.
 Let X be any uncountable set, such as ℝ or C and τ := {∅}∪{Y ⊂ X: X\Y is countable}. This is called the cocountable topology.
 Let X be any infinite set and τ := {∅}∪{Y ⊂ X: X\Y finite}. This is called the cofinite topology. If X = R or C then this is known as the Zariski toplogy.
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