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Let $X$ be a topological space and let $Y$ be a Hausdorff space. Let $D$ be dense in $X$. Prove that continuous functions $f, g : X \to Y$ which are equal in $D$ are equal in all $X$.

I'm a little stuck with this elementary proof. All help appreciated :)

Martin Sleziak
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Matias Heikkilä
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  • See http://math.stackexchange.com/questions/543962/f-g-continuous-from-x-to-y-if-they-are-agree-on-a-dense-set-a-of-x-th and maybe also http://math.stackexchange.com/questions/505/can-there-be-two-distinct-continuous-functions-that-are-equal-at-all-rationals – Martin Sleziak Dec 29 '14 at 08:24

1 Answers1

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Assume there is $x \in X$ such that $f(x) \neq g(x)$. Since $Y$ is Hausdorff, there are neighborhoods $U$ and $V$ of $f(x)$ and $g(x)$ respectively such that $U \cap V = \emptyset$. Since $f$ and $g$ are continuous, $f^{-1}(U) \cap g^{-1}(V)$ is open and nonempty (it contains $x$) in $X$ and therefore contains an element $x_0$ of $D$. Now, $f(x_0) \in U$ and, since $x_0 \in D$, $f(x_0) = g(x_0) \in V$, so $f(x_0) \in U \cap V$, contradicting the fact that $U$ and $V$ are disjoint.

Austin Rochford
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