Some theorems in general topology

Some theorems in general topology
Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine-Borel theorem).
Every continuous image of a compact space is compact.
Tychonoff's theorem: The (arbitrary) product of compact spaces is compact.
A compact subspace of a Hausdorff space is closed.
Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
Every sequence of points in a compact metric space has a convergent subsequence.
Every interval in R is connected.
Every compact m-manifold can be embedded in some Euclidean space Rn.
The continuous image of a connected space is connected.
A metric space is Hausdorff, also normal and paracompact.
The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
Any open subspace of a Baire space is itself a Baire space.
The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
General topology also has some surprising connections to other areas of mathematics. For example:
In number theory, Furstenberg's proof of the infinitude of primes.

- from http://en.wikipedia.org/wiki/Topology

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