Motivation

Many proofs are based on the same key ideas. It is good to isolate the key ideas in as general form as possible and then derive results there.

Definition

A Topological Space (X,τ) consists of a set X together with a collection τ of subsets of X, satisfying:

• 1) For any collection of elements of τ, the union of them is in τ.
• 2) For any finite collection of elements of τ is in τ.
• 3) The entire set X and the empty set are members of τ.

τ is called topology. Subsets of X (elements of τ) are called open sets.

Trivial Cases

Any set X can be made into a topological space by taking either (discrete topology) or by taking τ={X,0} (the indiscrete topology).

Interesting Cases

Let X=R (the set of real numbers) and let τ=all subsets of R which can be expressed as unions of open intervals (a,b).