Topology

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 τ=2^X (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).

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .