I haven't read the definition of local homeomorphism anywhere, but I would disagree with the definition given in the article: the first-coordinate inclusion R→R2 does not feel like a local homeomorphism to me.
You are certainly right. I rewrote the article: I think it looks a bit better now. Note that a surjective local homeomorphism between compact Hausdorff spaces is always a covering map but the converse fails, of course. --PST 20:58, 18 January 2009 (UTC)
"Every homeomorphism is of course also a local homeomorphism, but this is boring." This is poorly worded: 1) The coordinating conjunction "but" sets up a contradiction/negation of a previous idea, but the follow "this is boring" is either a positive addition to the point or a logical continuation (usually "and" or "so," respectively); and 2) "of course" and "boring" are hardly professional and only add to the verboseness of the statement. A better way to word this statement would be: "By definition, every homeomorphism is a local homeomorphism." This is clear and to the point and informs the reader of the fact at hand without extraneous language. — Preceding unsigned comment added by 220.127.116.11 (talk) 17:34, 17 March 2013 (UTC)
In the "Formal Definition" the open set U gets silently promoted to a topological space - one should mention that the subspace topology is assumed, which might not be obvious to someone approaching the definition for the first time.