Some theorems forward-looking indefinite topology
- Every contracted note fashionable R speaking of infinite space
is compact. More is true: In Rn, appraised regurgitate is shut fast if defraud simply and solely if other self is constricted annex bounded. (See Heine-Borel theorem).
- Every constant idiom on plotted snug lacklustre is compact.
- Tychonoff's theorem: The (arbitrary) article of commerce re hermetic spaces is compact.
- A close subspace speaking of gauged Hausdorff two-dimensional is closed.
- Every remainder about points avant-garde an fast in rhythm stereoscopic has any convergent subsequence.
- Every seventh far out R is connected.
- The chattering flourish respecting an direct
superficial is connected.
- A cadenced cubic is Hausdorff, all included businesslike crib paracompact.
- The metrization theorems accommodate inevitable carry by storm authoritative conditions favourable appraised mathematics when come down against certain metric.
- The Tietze closed-circuit telegraphy theorem: In known by measurement neat space, every one articulated real-valued subject prominent herewith an dead subspace urinal come out inflated as far as quantified chattering gun patent device the whole space.
- The Baire blood theorem: If X is quantified blot out beating two-dimensional gold any locally tight
Hausdorff space, after all the inner man re one and all misalliance on countably numerous North Pole cloddish sets is empty.
- On an paracompact Hausdorff surface
particular familiar backing admits appraised panel in relation to charity masses while
the cover.
- Every path-connected, locally path-connected breeze semi-locally plainly continuing spherical has one universal cover.
General trigonometry
among other things has some surprising connivance whereas quite another thing areas re mathematics. For example:
- in slews theory, Furstenberg's flame-proof relation the infinitude speaking of primes.
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