Manifold and Differential Forms

Abstract

Definition: A pair {X, τ} where X is an arbitrary set and τ a collection of subsets τi ∈ X is a topological space if τ satisfies the following conditions: 1) ⊘ ∈τ,X ∈ τ 2) If U 1 ∈ τ and U 2 ∈ τ,then U 1 ∩ U 2 ∈ τ 3) If U s ∈ τ for each s ∈ S where S is an arbitrary index set, then ∪U s ∈ τ Every subset U ⊂ X belonging to the collection τ is called an open set and the collection τ a topology in the set X. A neighbourhood of an element x ∈ X is an arbitrary open set containing x