(I have deleted my original answer to this question because the question was changed in such a way that it made my answer irrelevant.  I still think that my basic point was valid, and so am posting a new answer to make that point again.  As my original post gained quite a few votes, I judge it not ethical to completely reword my answer but keep those votes.  I am also making this answer "community wiki" not because I think anyone else should edit it but to remove it from the reputation/vote game.)

I think that the basic answer to this question is that there are connections between algebraic and topological things **because we look for them**.  And we look for them because we have, in the past, found them useful.  Something I continually (and I mean "continually", just ask one) tell my students is that mathematicians are fundamentally lazy.  If we have a good theorem, we don't just use it for what it was first proved for, we look for other ways to use it, ways to extend it, ways to push it further than it was ever intended to be pushed.  So if, as a topologist, I see the algebraists doing wonderful things with classifying and studying rings, then I'll do my best to make a ring out of my topological space so that I can steal (sorry, "use") their ideas and save myself a lot of bother.  Thus: cohomology theory and the whole area of homotopy theory.

That the reverse is true is no surprise.  Again, mathematicians are lazy so if we see a bridge with lots of useful stuff going in one direction, we ignore the "one way" signs and go the other way.

You could then ask "Why do the bridges exist at all?".  Well, they don't always exist.  Sometimes we can construct them and sometimes not.  It feels a bit like you are looking at a bridge and say "Wow!  Who would ever have thought of putting a bridge there?!" but ignoring all the stumps and collapsed half-bridges that litter the riverbank.  Of course, one can ask about a _specific_ bridge and ask why that one didn't collapse, but the question feels much more general than that.

So, in conclusion, that bridges exist is, I feel, more down to the downright mulishness of mathematicians determined to build a bridge wherever they can, regardless of how many collapses and "Pont d'Avignon"s they create in the process.

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The above, clearly, works for any two areas of mathematics.  Thinking particular of topological spaces, then I think that the questioner is missing the point of "near" and "far" a little when he says:

> However, when you start with algebraic objects and then get topological spaces out of them --- I find that surprising somehow because a priori there is not necessarily anything "geometric" or "topological" or "shape-y" or "neighborhood-y" going on.

(I should point out that the "near" and "far" bit added in the question is in response to my original answer.)

Consider this scenario:

1. I have something I don't know anything about.
2. Can I find something out about something **like it** but simpler?
3. Yes! Great!  But how do I **measure** which things are better **approximations** of my unknown thing than others?

Isn't that just what is going on in studying these algebraic objects?  The language is fundamentally topological so there's no surprise at all that topological spaces result.

So as soon as an area of mathematics becomes interesting in that there are things that you can't figure out easily and simply then the question of finding enough approximations comes in and thus topology.

In conclusion of this second part, "interesting = topological" so making the (bizarre) assumption that algebra is interesting, it must thus be topological.