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  • More on Convergence of Continuous Functions and Topological Convergence of Sets

  • Gerald Beer
  • Journal:
    Canadian Mathematical Bulletin / Volume 28 / Issue 1 / 01 March 1985
    Published online by Cambridge University Press:
    20 November 2018, pp. 52-59
    Print publication:
    01 March 1985
    • Article
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  • Let C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.