Topologies and all that -- A Tutorial

Abstract

This is a brief introduction to the basic concepts of topology. It includes the basic constructions, discusses separation properties, metric and pseudometric spaces, and gives some applications arising from the use of topology in computing.Proposition 1.1 A family β of sets is the base for a topology onKelley [Kel55, p. 47] gives the following example: Put X := {0, 1, 2}, A := {0, 1} and B := {1, 2}, then β := {X, A, B, ∅} cannot be the base for a topology. Assume it is, then the topology must be β itself, but A ∩ B ∈ β. So we have to be a bit careful. Let us have a look at the proof.Proof Checking the properties for a base shows that the condition is certainly necessary. Suppose that the condition holds, and defineThen ∅, X ∈ τ , and τ is closed under arbitrary unions, so that we have to check whether τ is closed under finite intersections. In fact, let x ∈ U ∩ V with U, V ∈ τ , then we can find U 0 , V 0 ∈ β with x ∈ U 0 ∩V 0 . By assumption there exists W ∈ β with x ∈ W ⊆ U 0 ∩V 0 ⊆ U ∩V , so that U ∩ V can be written as union of elements in β. ⊣