List questions from examples-counterexamples

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List Question

Connected metric spaces with at least 2 points are uncountable.

Complete space as a disjoint countable union of closed sets

Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

Relation between Fourier transform and Fourier series

If $M_0$, $M_1$, and $M_2$ are least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f''(x)|$, does $M_1^2\leq 4M_0M_2$ for vector valued functions?

Looking for examples of first countable, compact spaces which is not separable

Discontinuous functions with closed graphs

List applications of sets & relations in science/business/tech that a highschooler can understand

Differentiability of a function, of its sections and of its components

Discontinuous linear functional

Spaces with equal homotopy groups but different homology groups?

a simple example of a non-continuous function whose graph is homeomorphic to the domain

An equivalent definition of a group

How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

Finite Product of Closed Maps Need Not Be Closed

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