B-1 - Subspaces (Hereditary (P)-Spaces)

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If X is a given topological space with a topology O (the collection of open subsets) and X′ is a subset of X and if we put O = { X’ ⋂ O : O ∈ O }, then O ′ satisfies the conditions to be a topology on X ′, that is, X′ is a topological space with the topology O ′. This topological space X′ is called a subspace of X and the topology O ′ is called the relative topology (the subspace topology) of O with respect to X′ . Sometimes, a closed subset (open subset) is called a closed subspace (open subspace) in view of subspaces. The chapter lists some facts that are direct consequences of the definitions of subspace—such as a subset E ′ of X’ is a closed (open) subset of the space X if and only if (iff) E′ = X’ ∩ E’ for some closed (open) subset E of the space X .