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
A topological space is called compactly generated – also called a “k-space”^{1} (Gale 1950, 1., following lectures by Hurewicz in 1948), “Kelley space” (Gabriel & Zisman 1967, III.4), or “kaonic space” (Postnikov 1982, p. 34) – if its topology is detected by the continuous images of compact Hausdorff spaces inside it.
As opposed to general topological spaces, compactly generated spaces form a cartesian closed category while still being general enough for most purposes of general topology, hence form a convenient category of topological spaces (Steenrod 1967) and as such have come to be commonly used in the foundations of algebraic topology and homotopy theory, especially in their modern guise as compactly generated weakly Hausdorff spaces, due to McCord 1969, Sec. 2.
A function $f\colon X \to Y$ between topological spaces is $k$-continuous if for all compact Hausdorff spaces $C$ and continuous functions $t\colon C \to X$ the composite $f \circ t\colon C \to Y$ is continuous.
The following conditions on a space $X$ are equivalent:
For all spaces $Y$ and all functions $f\colon X \to Y$, $f$ is continuous if and only if $f$ is $k$-continuous.
There is a set $S$ of compact Hausdorff spaces such that the previous condition holds for all $C \in S$.
$X$ is an identification space of a disjoint union of compact Hausdorff spaces.
A subspace $U \subseteq X$ is open if and only if the preimage $t^{-1}(U)$ is open for any compact Hausdorff space $C$ and continuous $t\colon C \to X$.
A space $X$ is a $k$-space if any (hence all) of the above conditions hold. Some authors also say that a $k$-space is compactly generated, while others reserve that term for a $k$-space which is also weak Hausdorff, meaning that the image of any $t\colon C\to X$ is closed (when $C$ is compact Hausdorff). Some authors (especially the early authors on the subject) go on to require a Hausdorff space, but this seems to be unnecessary.
Examples of compactly generated spaces include
Every compact space is compactly generated.
Every locally compact space is compactly generated.
Every topological manifold is compactly generated
Every CW-complex is a compactly generated topological space.
Since a CW-complex $X$ is a colimit in Top over attachments of standard n-disks $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (prop.) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.
For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.
Every first countable space is a compactly generated space.
Since the topology is determined by convergent sequences = maps from one-point compactification $\mathbb{N} \cup \{\infty\}$); these include all Fréchet spaces.
Let $k\Top$ denote the category of $k$-spaces and continuous maps, and $\Top_k$ denote the category of all topological spaces and $k$-continuous maps. We have inclusions
of which the first is the inclusion of a full coreflective subcategory, the second is bijective on objects, and the composite $k\Top \to Top_k$ is an equivalence of categories.
The coreflection $\Top \to k\Top$ is usually denoted $k$ and sometimes called $k$-ification (May 1999, p. 49).
This functor is constructed as follows: we take $k(X)=X$ as a set, but with the topology whose closed sets are those whose intersection with compact Hausdorff subsets of (the original topology on) $X$ is closed (in the original topology on $X$). Then $k(X)$ has all the same closed sets and possibly more, hence all the same open sets and possibly more.
In particular, the identity map $id:k(X)\to X$ is continuous, and forms the counit of the coreflection. Thus this coreflection has a counit which is both monic and epic, i.e. a “bimorphism”—such a coreflection is sometimes called a “bicoreflection.”
Moreover, the identity $id: X \to k(X)$ is $k$-continuous, so that the counit becomes an isomorphism in $\Top_k$. This shows that $k\Top \to \Top_k$ is essentially surjective, and it is fully faithful since any $k$-continuous function between $k$-spaces is $k$-continuous; hence it is an equivalence.
Since $k\Top \hookrightarrow \Top$ is coreflective, it follows that $k\Top$ is complete and cocomplete. Its colimits are constructed as in $Top$, but its limits are the $k$-ification of limits in $Top$. This is nontrivial already for products: the $k$-space product $X\times Y$ is the $k$-ification of the usual product topology. The $k$-space product is better behaved in many ways; e.g. it enables geometric realization to preserve products (and all finite limits), and the product of two CW complexes to be another CW complex.
If one is interested in $k$-spaces which are also weak Hausdorff, then there is a further reflector which must be applied; see weakly Hausdorff space.
The categories $k\Top\simeq \Top_k$ are cartesian closed. (While in Top only some objects are exponentiable, see exponential law for spaces.) For arbitrary spaces $X$ and $Y$, define the test-open or compact-open topology on $\Top_k(X,Y)$ to have the subbase of sets $M(t,U)$, for a given compact Hausdorff space $C$, a map $t\colon C \to X$, and an open set $U$ in $Y$, where $M(t,U)$ consists of all $k$-continuous functions $f\colon X \to Y$ such that $f(t(C))\subseteq U$.
(This is slightly different from the usual compact-open topology if $X$ happens to have non-Hausdorff compact subspaces; in that case the usual definition includes such subspaces as tests, while the above definition excludes them. Of course, if $X$ itself is Hausdorff, then the two become identical.)
With this topology, $\Top_k(X,Y)$ becomes an exponential object in $Top_k$. It follows, by Yoneda lemma arguments (prop.), that the bijection
is actually an isomorphism in $\Top_k$, which we may call a $k$-homeomorphism (e.g. Strickland 09, prop. 2.12). In fact, it is actually a homeomorphism, i.e. an isomorphism already in $Top$.
It follows that the category $k\Top$ of $k$-spaces and continuous maps is also cartesian closed, since it is equivalent to $\Top_k$. Its exponential object is the $k$-ification of the one constructed above for $\Top_k$. Since for $k$-spaces, $k$-continuous implies continuous, the underlying set of this exponential space $k\Top(X,Y)$ is the set of all continuous maps from $X$ to $Y$. Thus, when $X$ is Hausdorff, we can identify this space with the $k$-ification of the usual compact-open topology on $Top(X,Y)$.
Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.
Unfortunately neither of the above categories is locally cartesian closed (Cagliari-Matovani-Vitale 95)
However, if $K$ is the category of not-necessarily-weak-Hausdorff k-spaces, and $A$ and $B$ are k-spaces that are weak Hausdorff, then the pullback functor $K/B\to K/A$ has a right adjoint. This is what May and Sigurdsson used in their book Parametrized homotopy theory.
There is still a lot of work on fibred exponential laws and their applications. One reason for the success and difficulties is that it is easy to give a topology on the space of closed subsets of a space $X$ by regarding this as the space of maps to the Sierpinski space (the set $\{0,1\}$ of truth values in which $\{1\}$ is closed but not open). From this one can get an exponential law for spaces over $B$ if $B$ is $T_0$, so that all fibres of spaces over $B$ are closed in their total space. Note that weak Hausdorff implies $T_0$.
The category of compactly generated Hausdorff spaces is a regular category. (Cagliari-Matovani-Vitale 95).
For every topological space $X$, the canonical continuous function from the $k$-ification (the adjunction counit) is a weak homotopy equivalence, hence induces an isomorphism on all homotopy groups:
The idea of compactly generated Hausdorff spaces first appears in print in:
where it is attributed to Witold Hurewicz, who introduced the concept in a lecture series given in Princeton, 1948-49, which Gale attended.^{2}
Early textbook accounts assuming the Hausdorff condition:
John Kelley, p. 230 in: General topology, D. van Nostrand, New York 1955, reprinted as: Graduate Texts in Mathematics, Springer 1955 (ISBN:978-0-387-90125-1)
James Dugundji, Section XI.9 of: Topology, Allyn and Bacon 1966 (pdf)
Pierre Gabriel, Michel Zisman, sections I.1.5.3 and III.2 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (pdf)
also:
Susan Niefield, Section 9 of: Cartesianness, PhD thesis, Rutgers 1978 (proquest:302920643)
Francis Borceux, Section 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)
Influential emphasis of the usefulness of the notion as providing a convenient category of topological spaces:
Early discussion in the context of geometric realization of simplicial topological spaces:
and briefly in
More history and early references, with emphasis on category-theoretic aspects:
The idea of generalizing compact generation to weakly Hausdorff spaces appears in:
where it is attributed to John C. Moore.
Review in this generality of CG weakly Hausdorff spaces:
Gaunce Lewis, Compactly generated spaces (pdf), appendix A of The Stable Category and Generalized Thom Spectra PhD thesis Chicago, 1978
Peter May, Chapter 5 of: A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)
Neil Strickland, The category of CGWH spaces, 2009 (pdf, pdf)
Charles Rezk, Compactly Generated Spaces, 2018 (pdf, pdf)
The terminology “kaonic spaces”, or rather the Russian version “каонные пространства” is used in
M M Postnikov, Введение в теорию Морса, Наука 1971 (web)
M M Postnikov, p. 34 of: Лекции по алгебраической топологии. Основы теории гомотопий, Наука 1982 (web)
Proof that k-spaces form a regular category:
Review with focus on compactly generated topological G-spaces in equivariant homotopy theory and specifically equivariant bundle-theory:
See also:
George W. Whitehead, Section I.4 of: Elements of Homotopy Theory, Springer 1978 (doi:10.1007/978-1-4612-6318-0)
Brian J. Day, Relationship of Spanier’s Quasi-topological Spaces to k-Spaces , M. Sc. thesis University of Sydney 1968. (pdf)
Peter Booth, Philip R. Heath, Renzo A. Piccinini, Fibre preserving maps and functional spaces, Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158–167, Lecture Notes in Math., 673, Springer, Berlin, 1978.
Yves Félix, Stephen Halperin, Jean-Claude Thomas, Section 0 of: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)
(in a context of rational homotopy theory)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, around note 4.3.22 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Martín Escardó, Jimmie Lawson, Alex Simpson, Comparing Cartesian closed categories of (core) compactly generated spaces, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 (doi:10.1016/j.topol.2004.02.011)
Ronnie Brown, Topology and groupoids, Booksurge 2006, section 5.9.
Samuel Smith, The homotopy theory of function spaces: a survey (arXiv:1009.0804)
Stefan Schwede, section A.2 of Symmetric spectra (2012)
The reason for choosing the term “k-space” in Gale 1950 seems to be lost in history. The “k” is not for “Kelley”, as Kelley 1955 came later. It might have been an allusion to the German word kompakt. ↩
This is according to personal communication by David Gale to William Lawvere in 2003, forwarded by Lawvere to Martin Escardo at that time, and then kindly forwarded by Escardo to the nForum in 2021; see there. ↩
Last revised on September 7, 2021 at 07:28:10. See the history of this page for a list of all contributions to it.