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I've studied the statement and proof of the Baire Category Theorem for complete metric spaces and locally compact Hausdorff spaces, and I've used the theorem in an exercise to prove that that in either of these types of spaces, every nonempty countable closed set contains an isolated point. However, I still don't have a firm grasp on what the uses of the Baire Category Theorem are. Had the above exercise not come with a hint to use the theorem, I wouldn't have known it would be useful.

When solving a problem in topology or analysis, are there any giveaways that the Baire Category Theorem is a good tool for the job? I'm lacking the intuition necessary to actually apply this theorem in practice.

Nick A.
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    I don't have time now to fully expand on this, but my intuition for using it to construct pathological objects is that you look for some kind of "approximate pathological behavior" that in a certain type of limiting sense reproduces the pathological behavior (e.g. finite decimal-expressed numbers approximate an irrational number, slopes greater than the positive integer $n$ approximate infinite slope, etc.) and such that (what follows relates to the last criteria for "nowhere dense" given in this MSE answer) (continued) – Dave L. Renfro Apr 25 '23 at 20:38
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    given an arbitrarily specified "approximate pathological behavior" and an arbitrary element in the ambient complete metric space and arbitrarily close to that element there exists an open ball all of whose elements satisfy the specified "approximate pathological behavior" (i.e. there exists arbitrarily close to that element another element that satisfies the specified "approximate pathological behavior" and such that all sufficiently small perturbations of that other element -- meaning all elements sufficiently close to it -- also satisfy the "approximate pathological behavior"). (continued) – Dave L. Renfro Apr 25 '23 at 20:47
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    With this in mind, see my first two comments to this mathoverflow question. Along with this, see the paragraph beginning with "An interesting feature of typical properties" in my answer to Generic Elements of a Set. For applications see the answers and comments to the mathoverflow question Classic applications of Baire category theorem. – Dave L. Renfro Apr 25 '23 at 20:47

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