topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
(maps from compact spaces to Hausdorff spaces are closed and proper)
Let be a continuous function between topological spaces such that
Then is
a closed map;
a proper map.
For the first statement, we need to show that if is a closed subset of , then also is a closed subset of .
Now
since closed subsets of compact spaces are compact it follows that is also compact;
since continuous images of compact spaces are compact it then follows that is compact;
since compact subspaces of Hausdorff spaces are closed it finally follow that is also closed in .
For the second statement we need to show that if is a compact subset, then also its pre-image is compact.
Now
since compact subspaces of Hausdorff spaces are closed it follows that is closed;
since pre-images under continuous of closed subsets are closed, also is closed;
since closed subsets of compact spaces are compact, it follows that is compact.
(continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms)
Let be a continuous function between topological spaces such that
Then is a homeomorphism, i. e. its inverse function is also a continuous function.
In particular then both and are compact Hausdorff spaces.
Write for the inverse function of .
We need to show that is continuous, hence that for an open subset, then also its pre-image is open in . By passage to complements, this is equivalent to the statement that for a closed subset then the pre-image is also closed in .
But since is the inverse function to , its pre-images are the images of . Hence the last statement above equivalently says that sends closed subsets to closed subsets. This is true by prop. .
The idea captured by corollary is that Hausdorffness is about having “enough” open sets whilst compactness is about having “not too many”. Thus a compact Hausdorff space has both “enough” and “not too many”. This theorem says that both conditions are at their limit: if we try to have more open sets, we lose compactness. If we try to have fewer open sets, we lose Hausdorffness.
Pontryagin's theorem establishes a bijection
between
the -Cohomotopy set of a closed (hence: compact) smooth manifold ;
the cobordism classes of its normally framed submanifolds of codimension ;
by taking the homotopy class of any map into the n-sphere to the preimage of any regular point (say , for definiteness) of a smooth representative :
To show that this construction is indeed injective, one needs that, similarly, the pre-image of any regular point of a smooth homotopy
between two such smooth representatives is a cobordism between and . But for a subspace of to constitute a cobordism between submanifolds of it is necessary that its projection onto the -factor is compact.
That this is implied here is guaranteed by the assumption that is closed, hence compact, so that also the product space is compact:
With this and since the n-sphere is, of course, Hausdorff, Prop. implies that the maps and above are all proper, hence that the corresponding pre-images (of singleton, hence compact, subspaces) are indeed compact, so that in particular the projection of the pre-image of to is compact (since continuous images of compact spaces are compact).
Last revised on February 3, 2021 at 23:28:40. See the history of this page for a list of all contributions to it.