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Suppose I have a map $f:X\rightarrow Y$ continuous, $X$ is compact, connected and also $f$ is a local homeomorphism, what condition should we include in $X$ so that $f$ becomes a covering map? Am I making any sense?

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    @mex you should list the defi$n$itio$n$ of the coveri$n$g map or give us a li$n$k i$n$ the question.2012-07-02

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If $X$ and $Y$ are Hausdorff, then compactness of $X$ and surjectivity of $f: X\to Y$ are already sufficient conditions for $f$ to be a covering map (edit: just to be clear, this is in addition to $f$ being a local homeomorphism). See this question.

Note that $f$ is necessarily surjective if $Y$ is assumed to be connected, because $f$ is closed and open (still under the hypothesis that $Y$ is Hausdorff).

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    @JuanS It was unclear to me, because the OP was already assuming that X was compact as well.2012-07-02