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Math Help - Tychonoff space

  1. #1
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    Tychonoff space

    I'm having difficulty with proving directly that every metric space is a Tychonoff space. I can show that every metric space is Hausdorff.
    Let (X,d) be a metric space, let A be a closed subset of X, let y \in X-A. Hint: Define f: X \rightarrow I by f(x)= min\{d(x,A)/d(y,A),1\}. So, f(y)=1 and f(x)=0 for every x \in A. I am stuck on proving that this function is continuous. Anyone can help me?
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  2. #2
    Super Member girdav's Avatar
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    Since if f and g are continuous then so is \min (f,g) you only have to show that x\mapsto d(x,A) is continuous. It's a consequence of the triangular inequality.
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  3. #3
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    Thank you very much for your help. I did prove x \mapsto d(x,A) continuous before. I guess now I just need to show that min \{f,g\} is continuous if f, g are continous, and this should be straightforward.
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