nLab
subsets are closed in a closed subspace precisely if they are closed in the ambient space

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Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

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topological homotopy theory

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Statement

Proposition

Let (X,τ)(X,\tau) be a topological space, and let CXC \subset X be a closed subset, regarded as a topological subspace (C,τ sub)(C,\tau_{sub}). Then a subset SCS \subset C is a closed subset of (C,τ sub)(C,\tau_{sub}) precisely if it is closed as a subset of (X,τ)(X,\tau).

Proof

If SCS \subset C is closed in (C,τ sub)(C,\tau_{sub}) this means equivalently that there is an open open subset VCV \subset C in (C,τ sub)(C, \tau_{sub}) such that

S=C\V. S = C \backslash V \,.

But by the definition of the subspace topology, this means equivalently that there is a subset UXU \subset X which is open in (X,τ)(X,\tau) such that V=UCV = U \cap C. Hence the above is equivalent to the existence of an open subset UXU \subset X such that

S =C\V =C\(UC) =C\U. \begin{aligned} S & = C \backslash V \\ & = C \backslash (U \cap C) \\ & = C \backslash U \end{aligned} \,.

But now the condition that CC itself is a closed subset of (X,τ)(X,\tau) means equivalently that there is an open subset WXW \subset X with C=X\WC = X \backslash W. Hence the above is equivalent to the existence of two open subsets W,UXW,U \subset X such that

S=(X\W)\U=X\(WU). S = (X \backslash W) \backslash U = X \backslash (W \cup U) \,.

Since the union WUW \cup U is again open, this implies that SS is closed in (X,τ)(X,\tau).

Conversely, that SXS \subset X is closed in (X,τ)(X,\tau) means that there exists an open TXT \subset X with S=X\TXS = X \backslash T \subset X. This means that S=SC=(X\T)C=C\T=C\(TC)S = S \cap C = (X \backslash T) \cap C = C\backslash T = C \backslash (T \cap C), and since TCT \cap C is open in (C,τ sub)(C,\tau_{sub}) by definition of the subspace topology, this means that SCS \subset C is closed in (C,τ sub)(C, \tau_{sub}).

Created on May 15, 2017 at 12:15:04. See the history of this page for a list of all contributions to it.