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
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A natural topology on mapping spaces of continuous functions, important because of its role in exhibiting locally compact topological spaces to be exponentiable, as demonstrated below, culminating in Corollary .
The compact-open topology on the set of continuous functions is generated by the subbasis of subsets that map a given compact subspace to a given open subset , whence the name.
When restricting to continuous functions between compactly generated topological spaces one usually modifies this definition to a subbase of open subsets , where now is the image of a compact topological space under any continuous function . This definition gives a cartesian internal hom in the category of compactly generated topological spaces (see also at convenient category of topological spaces).
The two definitions agree when the domain is a compactly generated Hausdorff space, but not in general. Beware that texts on compactly generated spaces nevertheless commonly say “compact-open topology” for the second definition.
Let and be topological spaces. We denote by the set of continuous maps from to .
Other common notations for are or .
Let be a topological space. We denote by the set of subsets of which are compact with respect to .
Let be a topological space. Let be a subset of . We denote by the topological closure of in with respect to .
A topological space is locally compact if the set of compact neighbourhoods of points of form a neighbourhood basis of . That is to say: for every and every neighbourhood of , there is a and a such that .
There are many variations on this definition, which can be found at locally compact space. These are all equivalent if is Hausdorff. We do not however make the assumption that is Hausdorff.
Let and be topological spaces. Given and , we denote by the set of continuous maps such that .
Let and be topological spaces. The compact-open topology on is that with sub-basis given by the set of sets such that and .
Let and be topological spaces. We shall denote the compact-open topology (def. ) on by .
Let be a locally compact topological space, def. , and let be a topological space. The map given by is continuous, where is equipped with the product topology with respect to and (notation ).
By an elementary fact concerning continuous maps, it suffices to show that for any , and any such that , there is a such that , and such that .
To demonstrate this, we make the following observations.
1) Since is continuous, we have that .
2) Since is locally compact, we deduce from 1) that there is a and a such that .
3) By 2) and by definition of and , we have that , and that .
4) Let , Since , and since , the latter inclusion holding by definition of , we have that . We deduce that .
By 3) and 4), we see that we can take as the beginning of the proof to be .
Let , , and be topological spaces. Let be a continuous map, where is equipped with the product topology with respect to and . Then the map given by is continuous, where is given by , and where is equipped with the compact-open topology .
By an elementary fact concerning continuous maps, it suffices to show that for any , and any such that , there is a such that , and such that .
To demonstrate this, we make the following observations.
1) Since , we have, by definition of and by definition of , that for all .
2) Since is continuous, we have that .
3) By 1), we have that .
4) Let denote the subspace topology on with respect to . By 2), we have that .
5) By 3), we have that .
6) Since , it follows from 4), 5), and the tube lemma that there is a such that and such that .
7) We deduce from 6) that . This is the same as to say that for all . Thus belongs to for all , which is the same as to say that .
We conclude that we can take the required of the beginning of the proof to be the of 6).
Let , , and be topological spaces. Suppose that is locally compact. Let be a continuous map, where is equipped with the compact-open topology . Then the map given by is continuous, where is equipped with the product topology .
We make the following observations.
1) We have that , where is given by .
2) By Proposition , we have that is continuous.
3) It is an elementary fact that is continuous.
4) Since is continuous, it follows by an elementary fact that is continuous.
5) We deduce from 2) - 4) that is continuous. By 1), we conclude that is continuous.
Let and be topological spaces. Suppose that is locally compact. Then together with the corresponding map defines an exponential object in the category Top of all topological spaces.
Follows immediately from Proposition , Proposition , and the fact that and are exhibited by the corresponding maps and of Proposition and Proposition to define an exponential object in the category of sets.
A proof can also be found in Aguilar-Gitler-Prieto 02, prop. 1.3.1, or just about any half-decent textbook on point-set topology!
However, the result is almost universally stated with an assumption that is Hausdorff which, as the proof we have given illustrates, is not needed.
We moreover have a homeomorphism if in addition is Hausdorff. See also convenient category of topological spaces.
(mapping space construction out of the point space is the identity)
The point space is clearly a locally compact topological space. Hence for every topological space the mapping space exists. This is homeomorphic to the space itself:
If is a metric space then the compact-open topology on is the topology of uniform convergence on compact subsets in the sense that in with the compact-open topology iff for every compact subset , uniformly on . If (in addition) the domain is compact then this is the topology of uniform convergence.
(loop space and path space)
Let be any topological space.
The standard circle is a compact Hausdorff space hence a locally compact topological space. Accordingly the mapping space
exists. This is called the free loop space of .
If both and are equipped with a choice of point (“basepoint”) , , then the topological subspace
on those functions which take the basepoint of to that of , is called the loop space of , or sometimes based loop space, for emphasis.
Similarly the closed interval is a compact Hausdorff space hence a locally compact topological space (def. ). Accordingly the mapping space
exists. Again if is equipped with a choice of basepoint , then the topological subspace of those functions that take to that chosen basepoint is called the path space of :
Notice that we may encode these subspaces more abstractly in terms of universal properties:
The path space and the loop space are characterized, up to homeomorphisms, as being the limiting cones in the following pullback diagrams of topological spaces:
Here on the right we are using that the mapping space construction is a functor and we are using example in the identification on the bottom right mapping space out of the point space.
Textbook accounts:
Glen Bredon, Section VII.2 of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)
Francis Borceux, Sections 7.1 & 7.2 of: Categories and Structures, Vol. 2 of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Sections 1.2, 1.3 of: Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
See also:
Wikipedia, Compact-open topology
Ralph H. Fox, On Topologies for Function Spaces, Bull. AMS 51 (1945) pp.429-432. (pdf)
Eva Lowen-Colebunders, Günther Richter, An Elementary Approach to Exponential Spaces, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)
Last revised on December 3, 2022 at 14:07:59. See the history of this page for a list of all contributions to it.