TITLE: Why does topology rarely come up outside of topology?
QUESTION [42 upvotes]: I am currently taking topology and it seems like a completely different branch of math than anything else I have encountered previously. 
I find it a little strange that things are not defined more concretely. For example, a topological space is defined as a set $X$ with a collection of open sets $\tau$ satisfying some properties such as the empty set and $X$ are in $\tau$, intersection of two open sets are in $\tau$, and unions of open sets is in $\tau$. 
So, it seems that a lot of things are topological spaces, such as the real line equipped with a collection of open sets. But I have not seen anyone bringing this up in other areas of mathematics such as linear algebra, calculus, differential equations or analysis or complex analysis. Sure, open sets and closed sets are brought up but the concept of "topology", "base", etc. etc. are missing entirely.
As you scratch the surface a little more you encounter things such as the subspace topology, product topology, order topology and open sets are defined differently with respect to each of them. But nonetheless outside of a course in topology, you never encounter these concepts.
Is there a reason why topology is not essential for other courses that I have mentioned? Is there a good reference that meshes serious topology (as in Munkres) with more applied area of mathematics?

REPLY [15 votes]: A major topic of classical analysis is Fourier series and Fourier integrals. These ideas generalize to analysis on (locally compact) topological groups. A topological group is a group $G$ in which group multiplication $G \times G \rightarrow G$ where $(g,h) \mapsto gh$ and group inversion $G \rightarrow G$ where $g \mapsto g^{-1}$ are both continuous (using the product topology on $G \times G$ in order to speak about a function on it being continuous). Analysis on topological groups is a major theme within representation theory. You might have heard about Fourier series in an undergraduate analysis class, but such a course would not have discussed the Fourier transform on topological groups because the audience wouldn't have the experience to appreciate such a generalization yet. It would look "too abstract."
In addition to topological groups there are topological vector spaces: vector spaces $V$ (over the real numbers, say) in which vector addition $V \times V \rightarrow V$ where $(v,w) \mapsto v+w$ and scalar multiplication $\mathbf R \times V \rightarrow V$ where $(c,v) \mapsto cv$ are continuous, using the product topology on both $V \times V$ and $\mathbf R \times V$ in order to speak about functions on them being continuous. A special feature of finite-dimensional real vector spaces like $V = \mathbf R^n$ is that the usual topology on them is the only topology they have that makes them Hausdorff topological vector spaces. (What about the discrete topology on $\mathbf R^n$? Vector addition on $\mathbf R^n$ is continuous when $\mathbf R^n$ has the discrete topology, but scalar multiplication $\mathbf R \times \mathbf R^n \rightarrow \mathbf R^n$ when $\mathbf R^n$ has the discrete topology and the scalars $\mathbf R$ have their usual topology is not continuous.) That $\mathbf R^n$ has only one Hausdorff topological vector space structure is in some sense why we can talk about concepts like continuity in multivariable calculus without having to get into a treatment of topology first: the usual way we think about continuity of functions on $\mathbf R^n$ is the only reasonable way to do so. However, once you pass to infinite-dimensional spaces the situation changes: these spaces can be made into topological vector spaces in more than one interesting way, and this quickly leads into the area of functional analysis, which is not something you would have seen yet just because you can't learn everything in your first two years of college.  Functional analysis is not just abstraction for the sake of pure math: it's the mathematical foundation of quantum physics.
The language of topology is relevant to areas of math that at first glance seem  to be unrelated to issues of continuity, such as number theory. The study of solutions to congruences mod $m$ can be reduced to the case that $m = p^k$ is a power of a prime number $p$, and the best way to think systematically about congruences  modulo prime powers uses the $p$-adic integers $\mathbf Z_p$ (a compact ring) and the $p$-adic numbers $\mathbf Q_p$ (a locally compact field). A buzzword to look up in this context is "Hensel's lemma," which is the $p$-adic analogue of Newton's method from classical analysis.  The $p$-adic numbers were created by Hensel at the end of the 19th century, but the original description of his ideas were muddled and awkward because he lacked the topological language that greatly simplifies what is going on (once you are used to the language so that you can recognize certain topological features of the situation). If you've never heard of $p$-adic numbers, to convey their importance I'll just point out that the solution to Fermat's Last Theorem depends on them: the work by Wiles is concerned with representations of Galois groups into $p$-adic matrix groups.
The construction of $p$-adic numbers can be extended to the notion of an inverse limit, leading to constructions such as profinite groups (an inverse limit of finite groups). To work with inverse limits you need the language of topology that you think is not used elsewhere but really is: product topology, subspace topology, and base.