Continuous Functions Preserve Limit Points

Last updated Nov 1, 2022

Let $(X, \tau), (Y, \rho)$ be Topological Spaces and suppose $f: X \to Y$ is Continuous Function. Suppose $S \subset X$. If $x \in X$ is a Limit Point of $S$, then $f(x)$ is a Limit Point of $f(S)$.

Let $T = f(S)$ and $y = f(x)$. Suppose $V \subset Y$ is Open and $y \in V$. Then $f^{-1}(V)$ is Open ($f$ is Continuous Function) in $X$ and $x \in f^{-1}(V)$ since $f(x) = y \in V$. Since $x$ is a Limit Point of $S$, we know $f^{-1}(V) \cap S \neq \emptyset$. Thus there exists $z \in S$ so that $f(z) \in V$. Since $z \in S$, we also see $f(z) \in f(S) = T$. Thus $V \cap T \supset {z}$. Since $V$ was arbitrary, we have that $y$ is a Limit Point of $T$. $\blacksquare$

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