Over on her (very new) weblog, Sarah ponders the Hawaiʻian earring space and the infinite bouquet of circles. She illustrates the earring well, but just to be clear about the latter…
Take one circle with a marked point for each natural number and quotient by an equivalence relation declaring that all those marked points are really the same point. And don’t try to think of it as taking place in any finite-dimensional space. The bouquet of circles has exactly the quotient topology, and not the topology induced by embedding into any finite-dimensional space (that I know of, at least).
Anyhow, a special case of this caveat was exactly what struck her — that the underlying point sets of the earring and the bouquet are clearly isomorphic, but that the topologies are not. In particular, the earring is compact, while the bouquet is not. I believe I’ve discussed enough topology for these two facts to be an exercise for anyone who’s followed everything I’ve said here.
In a comment there, I wave my hands towards one example of where this difference shows up, and I’ll make you go give her some page-views to read it. Show the love!