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
The point space is the topological space whose underlying set is the singleton, and equipped with the unique topology that this set carries.
The point space is the terminal object in the category Top of topological spaces.
For any topological space, then for every element of its underlying set there is a continuous function from the point space
whose image is that point, and every such continuous function arises this way
For the following we write the point space explicitly as
For a topological space, then there is a bijection between the irreducible closed subspaces of and the frame homomorphisms from to from the frame of opens of to that of the point space. Moreover, this is given by
where is the union of all elements such that :
See also (Johnstone 82, II 1.3).
First we need to show that the function is well defined in that given a frame homomorphism then is indeed an irreducible closed subspace.
To that end observe that:
If there are two elements with then or .
This is because
where the first equality holds because preserves finite intersections by def. , the inclusion holds because respects inclusions by remark , and the second equality holds because preserves arbitrary unions by def. . But in the intersection of two open subsets is empty precisely if at least one of them is empty, hence or . But this means that or , as claimed.
Now according to prop. the condition identifies the complement as an irreducible closed subspace of .
Conversely, given an irreducible closed subset , define by
This does preserve
arbitrary unions
because precisely if which is the case precisely if all , which means that all and because ;
while as soon as one of the is not contained in , which means that one of the which means that ;
finite intersections
because if , then by or , whence or , whence with also ;
while if is not contained in then neither nor is contained in and hence with also .
Hence this is indeed a frame homomorphism .
Finally, it is clear that these two operations are inverse to each other.
examples of universal constructions of topological spaces:
Last revised on May 9, 2017 at 07:52:52. See the history of this page for a list of all contributions to it.