Tag Archives: product topology

Topology: Connected Spaces

Let X be a topological space. Recall that if U is a clopen (i.e. open and closed) subset of X, then X is the topological disjoint union of U and X–U. Hence, if we assume X cannot be decomposed any further, there’re no non-trivial clopen subsets of X. … Continue reading

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Topology: Product Spaces (II)

The Box Topology Following an earlier article on products of two topological spaces, we’ll now talk about a product of possibly infinitely many topological spaces. Suppose is a collection of topological spaces indexed by I, and we wish to define … Continue reading

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Topology: Interior

Let Y be a subset of a topological space X. In the previous article, we defined the closure of Y as the smallest closed subset of X containing Y. Dually, we shall now define the interior of Y to be the largest open subset contained in … Continue reading

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Topology: Closure

Suppose Y is a subset of a topological space X. We define cl(Y) to be the “smallest” closed subset containing Y. Its formal definition is as follows. Let Σ be the collection of all closed subsets containing Y. Note that , so Σ is not empty. … Continue reading

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Topology: Cauchy Sequences and Uniform Continuity

 [ Updated on 8 Mar 13 to include Cauchy-continuity and added answers to exercises. ] We wish to generalise the concept of Cauchy sequences to metric spaces. Recall that on an intuitive level, a Cauchy sequence is one where the … Continue reading

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Topology: Continuous Maps

Continuity in Metric Spaces Following the case of real analysis, let’s define continuous functions via the usual ε-δ definition. Definition. Let (X, d) and (Y, d’) be two metric spaces. A function f : X → Y is said to be … Continue reading

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Topology: Disjoint Unions

Disjoint Unions Let X and Y be topological spaces and be a set-theoretic disjoint union. We wish to define a topology on Z in a most natural way. Definition. The topology on is defined to be: It’s almost trivial to check that this … Continue reading

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