Topology is your friend!
ehrenfest said:
How useful is topology in theoretical physics?
Topology is an essential core topic of mathematics, and physical discourse cannot proceed without constant appeals mathematical reasoning.
ehrenfest said:
By topology, I mean the contents of Munkres book, Hausdorff spaces, homeomorphisms, etc. It seems to me like topology is totally a mathematical construct since the idea of an "open set" in an abstract space seems to have no "physical" meaning outside of Euclidean spaces. So do any of you theoretical physics actually use what is in Munkres?
Topology has many inspirations, but it might be useful to compare and contrast just two:
- much of topology (particularly geometric topology and much algebraic topology) concerns the topology of manifolds, including finite dimensional Lie groups.
- much of the rest of topology (including many topics in the book by Munkres, including the Tychonoff product theorem) is motivated by considering function spaces , probability spaces, etc., as topological spaces (typically infinite dimensional).
I hasten to add that these divisions are not hard and fast; topology is a unified subject and the interconnections between topology of finite dimensional manifolds and topology of large function spaces are many and intricate.
(Mathematical adventurers will be intrigued to learn that much topology nowadays is motivated by
logic! And I for one foresee applications to dynamical systems in this work.)
ehrenfest said:
BTW: I am trying to decide whether to take the semester of a topology course and I am really not wanting to for the reasons above.
If you really master this stuff you'll always be grateful for the opportunity to take a solid course in "general topology". A solid grounding will make it much easier to appreciate topological nuances in analysis and the modern theory of differential equations (functional analysis, operator theory, measure and probability theory are all needed for quantum mechanics, ergodic theory, and other core topics).
ehrenfest said:
I mean give me a physical example of a "one-point compactification"
The one point compactification of the plane is ... (fill in the blank). This came up just the other day in my remarks on stereographic projection.
[EDIT: And it came up just the other day elsewhere; see
this query.]
ehrenfest said:
or when you would use the "Tychanoff Theorem"
Tychonoff.
One of the most useful theorems in mathematics! "Nice" metric spaces tend to be compact; more generally, "nice" Hausdorff spaces tend to be compact. But large function spaces typically are
not compact. (As one punning slogan has it, "Nothing finer than a CH space!"
) If you look through a bunch of analysis and topology textbooks and write down the proofs of all the theorems whose proofs mention it (good exercise, BTW!), you'll see that one reason the Tychonoff theorem is so usefull is it guarantees
compactness.
Some pointers:
- Haar measure is needed for all kinds of applications of Lie groups in physics,
- Riesz Representation Theorem, one of the core results in functional analysis.
- Gelfand Theorem, a core result in the theory of C-* algebras (the foundation of geometric quantization; see this introduction by master expositor John Baez) and noncommutive topology; the basic idea here is to generalize the duality between statements about LCH spaces X and statements about the semisimple commutative Banach algebra of continuous asymptotically vanishing functions on X (this program has far-reaching implications for physics).
ehrenfest said:
or where I can find a locally compact Hausdorff space that is not metric? It seems like mathematicians came up with some of this stuff just out of sheer boredom.
If you look through a bunch of analysis and topology textbooks and write down the statements of all the theorems which mention "locally compact Hausdorff (LCH) spaces" (good exercise, BTW!), you'll see that one reason why (fill in the blank for the appropriate notion of morphism) among LCH spaces form such an important category is that it enjoys good "closure" properties, and offers many of the benefits which accrue from compactness.
For an example of an LCH space which is not a compact metric space, you can search the
Questions in Topology from the
Topology Atlas, where you can
Ask a Topologist 
See also Lynn Arthur Steen and J. Arthur Seebach, Jr,
Counterexamples in Topology, Springer, 1978, for zillions of counterexamples.
Trust me, stuff you see in mainstream textbooks in core topics like topology almost
never has the character of a "fantasy invented out of boredom", it is there (chosen from many competing topics in this huge, huge, huge field) because it has proven extremely useful in a great variety of disciplines. For example,
harmonic analysis unifies large swathes of
Fourier analysis,
representation theory, and
invariant theory, and is essential in many parts of theoretical physics. (Just look for physics eprints mentioning vector and tensor harmonics, for example! And that's just the trivial stuff!)
[EDIT: From
Mathworld:
Willard (1970: a compact Hausdorff space X is metrizable iff \left{ x, x \in X \right} = f^{-1}(0) for some continuous function.]
