Topologies and Sheaves

Abstract

Let X be a topological space, and let T denote the family of all open subsets of X. T becomes a category if we define

$$ Hom(U,V) = \left\{ {\begin{array}{*{20}{c}} 0&{if U \not\subset V} \\ {inclusion U \to V}&{if U \subset V} \end{array}} \right.$$

for U,V ∈ T. X is a final object in the category T. The intersection ∩U _i of finitely many U₁,…, U _n in T is equal to the product of the U₁,…, U _n in the category T. The union ∪ U _i of arbitrarily many U _i in T is equal to the direct sum of the U _i in the category T (cp. 0.1.1).