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In the chapter on Covering Spaces in his book A Basic Course in Algebraic Topology Massey uses the term topologically onto when defining covering spaces (see below for Massey's definition). What does it mean for a map to be topologically onto? The only thing I found using google is Definition 2.2. in http://www.emis.de/journals/HOA/IJMMS/2004/65-683717.pdf. Is that the standard definition? Does Massey's definition agree with the one you can find on Wikipedia?

Here is Massey's Definition of a Covering Space (he assumes all spaces involved to be path-connected and locally path-connected).

Definition. A covering space of a topological space $X$ is a continuous map $p: \tilde{X} \to X$ such that the following condition holds: Each point $x \in X$ has an arcwise-connected open neighbourhood $U$ such that each arc component of $p^{-1}(U)$ is mapped topologically onto $U$ by $p$.

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    Ok, so it is simply the definition found on Wikipedia plus some extra assumptions about path-connectedness. Thank you!2012-06-04

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