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Suppose $X$ and $Y$ are topological spaces and $f : X \to Y$ is a continuous, surjective map. Let $Y$ be connected.

Pick $y \in Y$ and suppose that $p^{-1}(y)$ is a finite set, with $|p^{-1}(y)| = n$. Let $V$ be a neighbourhood of $y$, is there a theorem that states that for any $y' \in V$, with $y' \neq y$ it follows that $|p^{-1}(y')| = n$?

Essentially what this is saying that the cardinality of the preimage is a locally constant function, I've never seen this result before, only something similar in a slightly different case (which dealt with compactness) in Topology from the Differentiable Viewpoint by John Milnor

Is there such a theorem, that the cardinality of the preimage is a locally constant function? If so how can I prove it? Also does a stronger version of this theorem hold, with fewer hypothesis (e.g. can we remove connectedness of $Y$)?

If a theorem doesn't exist with the current hypotheses, what hypotheses are needed to be added to arrive at the desired result?

Perturbative
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    The statement isn't true as written: Take $X = [-1, 1]$, $Y = [0, 1]$ both with the subspace topology from $\mathbb{R}$. Then $f:X \to Y$, $x \mapsto x^2$ is continuous and surjective, but $|f^{-1}(0)| = 1$. For any neighbourhood of $y$ however, there is some $y' \neq 0$, with $f(\sqrt{y'}) = f(-\sqrt{y'})=y'$, so $|f^{-1}(y)| = 2$. – B. Mehta Nov 29 '17 at 00:01
  • It might be worth pointing out this condition vaguely resembles that of being evenly covered in algebraic topology. – B. Mehta Nov 29 '17 at 00:03
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    Related: https://math.stackexchange.com/questions/807532/local-homeomorphism-and-inverse-image – symplectomorphic Nov 29 '17 at 00:32

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