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
Given a topological space , a closed subspace of is irreducible if it is inhabited and not the union of two closed proper (i.e. smaller) subspaces. In other words, is irreducible if whenever and are closed subsets of such that
then or .
Equivalently this may be expressed in terms of open subsets:
Let be a topological space, and let be a proper open subset, so that the complement is an inhabited closed subspace. Then is irreducible in the sense of def. precisely if whenever are open subsets with then or :
Every closed subset may be exhibited as the complement
for some open subset . Observe that under this identification the condition that is equivalent to the condition that , because it is equivalent to the equation labeled in the following sequence of equations:
Similarly, the condition that is equivalent to the condition that , because it is equivalent to the equality in the following sequence of equalities:
Under these equivalences, the two conditions are manifestly the same.
Yet another equivalent characterization is in terms of frame homomorphisms:
In the following we write
for the point, regarded, uniquely, as a topological space, the point space.
For a topological space, then there is a bijection between the irreducible closed subspaces of and the frame homomorphisms from to , 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, the inclusion holds because respects inclusions, and the second equality holds because preserves arbitrary unions). 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.
Note that frame homomorphisms from to are the same thing as continuous maps from the locale to the locale (that is, in the other direction, regarding the frames of opens as instead being locales). Thus, the irreducible closed subspaces correspond to what should be the points of the space, if we regard it as much as possible as a locale, that is the points of the locale . At a more elementary level, these are also the same thing as the completely prime filters in the frame .
Note that the closure of (the singleton set on) any point/element of is an irreducible closed subspace. is sober if and only if every irreducible closed subspace is the closure of a unique point of . In general, the irreducible closed subspaces of correspond to the points of the topological locale , which are (by definition) the completely prime filters on the frame of open subspaces of . Specifically, given an irreducibly closed subspace, the filter of open subspaces that contain it is completely prime; conversely, given a completely prime filter of open subspaces, the closure of its intersection is irreducible.
The theory of irreducible closed subspaces is not useful in constructive mathematics; instead, one must use the completely prime filters directly. While one might hope that the irreducibly open subsets (that is those that satisfy the conditions of Proposition ) might be more tractible constructively, they are in fact no better. (We should probably put in the classical proof and see where it goes wrong.)
Last revised on April 24, 2017 at 19:37:23. See the history of this page for a list of all contributions to it.