This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $\frac{1}{n}$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere
Be good at coming up with counter example in Topology
examples-counterexamplesgeneral-topologyintuition
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Your proof is not correct as it stands, as $U_a$ is not equal to $\{a\}$ but we know that $U_a \cap A \subseteq \{a\}$, in general.
We also don't really need the fact that $A$ is closed (though this is true). Pick for every $x \in X$ some open $U_x$ that contains $x$ and such that $U_x \cap A \subseteq \{x\}$. This is, by definition almost, an open cover of $X$. Compactness tells us that there is a finite subset $F$ of $X$ such that $X = \cup_{x \in F} U_x$. But then $A = X \cap A = A \cap (\cup_{x \in F} U_x) = \cup_{x \in F} (U_x \cap A) \subseteq F$, by how the $U_x$ were chosen, and this contradicts that $A$ is infinite.
A countably compact space that is not compact is the first uncountable ordinal, $\omega_1$, in the order topology. Or $\{0,1\}^\mathbb{R} \setminus \{\underline{0}\}$ and many more.
You are right that for metric spaces the reverse does hold, but this is atypical.
Here is a quick graph and how $d(x,y)$ works. Notice that near $0$, $d(x,y)$ evaluates the distance of $x,y$ as if it were the absolute value, whereas for large $|x|$ things are completely different: distances are quite small. In fact the "biggest" distance you'd get is $\pi$ (and is never attained of course).
Best Answer
I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.