Spaces with this property are conceptually the topological analogue of a finite set in set theory. For 10 points each:
[10m] Name this ubiquitously discussed property in analysis and topology held by subsets of Rn (“R-N”) that are closed and bounded.
ANSWER: compactness [or compact spaces or compact sets]
[10h] This theorem states that an equicontinuous and bounded real-valued function on a compact set converges uniformly. It is often proven by taking a countable dense subset of a space, then using the Bolzano–Weierstrass theorem, then diagonalizing to construct a convergent subsequence.
ANSWER: Arzela–Ascoli theorem [or Ascoli–Arzela theorem]
[10e] Uniform convergence holds if the difference between terms becomes arbitrarily small for some N that depends only on a quantity denoted by this letter. Continuity and convergence are formalized by a definition named for this letter and delta.
ANSWER: epsilon [or epsilon–delta definition]
<UBC A, Other Science>