# Definition of topologically onto map and Covering Spaces

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 $p$.

• I think it means "homeomorphism onto image". – Zhen Lin Jun 4 '12 at 8:55
• Zhen is right; here "mapped topologically" means "mapped homeomorphically." In other words, for each component $C$ of $p^{-1}(U)$, the restriction $C\to U$ of $p$ is a homeomorphism. – Jeremy Brazas Jun 4 '12 at 13:06
• Ok, so it is simply the definition found on Wikipedia plus some extra assumptions about path-connectedness. Thank you! – Lennart Jun 4 '12 at 14:03