Viewing topic: Imbedding Theorem

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Imbedding Theorem

  • Definition: Let X be a space in which one-point sets are closed. Suppose that {f_a}_(a in J) is an indexed family of continuous functions f_a: X -> R satisfying the requirement that for each point x_0 of X and each neighborhood U of x_0, there is an index a such that f_a is positive at x_0 and vanishes outside U. Then the function F: X -> R^J defined by F(x) =(f_a(x))_(a in J) is an imbedding of X in R^J. If f)a maps X into [0, 1] for each a, then F imbeds X in [0, 1]^J.

    Source: Topology (second edition) by James R. Munkres

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