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Math Help - Continuity on hausdorff spaces

  1. #1
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    Continuity on hausdorff spaces

    Let f and g be two continous functions on a topological space X with values on a hausdorff space Y. Prove that the set \{x \in X : f(x)=g(x)\} is closed in X.

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  2. #2
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    Quote Originally Posted by Inti View Post
    Let f and g be two continous functions on a topological space X with values on a Hausdorff space Y. Prove that the set \{x \in X : f(x)=g(x)\} is closed in X.
    Suppose that z\notin\{x \in X : f(x)=g(x)\}.
    Because Y is Hausdroff then <br />
\left( {\exists O_z } \right)\left[ {f(z) \in O_z } \right],\;\left( {\exists Q_z } \right)\left[ {g(z) \in Q_z } \right]\;\& \;O_z  \cap Q_z  = \emptyset .

    Because f~\&~g are continuous then z \in f^{ - 1} \left( {O_z } \right) \cap g^{ - 1} \left( {Q_z } \right) their intersection is open.
    Clearly t \in f^{ - 1} \left( {O_z } \right) \cap g^{ - 1} \left( {Q_z } \right)\; \Rightarrow \;f(t) \ne g(t).

    Does this show that the complement of \{x \in X : f(x)=g(x)\} is open?
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