topology (point-set topology, point-free topology)
see also 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
Topological spaces are very useful, but also admit many pathologies. (Although it should be admitted that often, one person’s pathology is another’s primary example.) Over the years, topologists have accumulated many different conditions to impose on topological spaces to exclude various spaces considered “pathological;” here we list some of the most important of these conditions.
Sometimes one also imposes these conditions to ensure better behavior of the resulting category of spaces; see nice category of spaces for more details, and also dichotomy between nice objects and nice categories. In many cases, a category of nice spaces will be reflective or coreflective.
There is a whole slew of separation conditions, of which the most common is the Hausdorff condition (any two points can be separated by disjoint opens). Hausdorff is also called $T_2$ and fits in a hierarchy ranging from $T_1$ through $T_4$ originally and now (at least) $T_0$ to $T_6$ (with some fractional subscripts too).
Sobriety is a separation condition living in between $T_0$ and $T_2$ (but incomparable with $T_1$), which guarantees a good relationship with locales.
A space is compact if any open cover of it has a finite subcover. Variations include locally compact, countably compact, sequentially compact, etc.
Locally compact Hausdorff spaces deserve special mention since they are exponentiable in Top.
A compactly generated space is, essentially, one whose topology is determined by its restriction to compact subspaces. These are notable because the category of compactly generated spaces is cartesian closed.
A metrizable space is one whose topology can be defined by a metric. We also have pseudometric spaces, quasimetric spaces, uniformizable spaces, etc.
A sequential space is one whose topology is determined by convergence of sequences. Note that any topology is determined by convergence of nets or filters.
A CW complex is a space built out of nothing by progressively attaching cells of higher and higher dimension. More generally, a cell complex is a space built by attaching cells, without regard to dimension (that is, lower-dimensional cells may be attached to higher-dimensional ones), and an m-cofibrant space is one that is homotopy equivalent to a CW complex (or equivalently a cell complex). These types of spaces are important for homotopy theory because they turn weak homotopy equivalences into homotopy equivalences.
A (topological) manifold is a space that is locally homeomorphic to $\mathbb{R}^n$ for some $n$.
(add your favorite!)
David Roberts: How about mentioning Alexander Grothendieck’s notion of tame topology? I saw in the video of Scharlau’s talk that G. wrote a lot on this, but the manuscript is lost. Do we have any idea past a vague description? (in Recoltes et Semailles or La longue marche I think.)
Todd Trimble: Quite a few people have thought long and hard about Grothendieck’s speculations on tame topology in his Esquisse d’un Programme. The approach with which I am most familiar comes from model theory, and falls under the rubric of “o-minimal structures” (the “o” standing for “order”). See the book Tame topology and o-minimal structures by van den Dries. A space which belongs to an o-minimal structure is a subspace of some Euclidean space $\mathbb{R}^n$ and turns out to be indeed nice (it admits nice triangulations for instance).
In a sense this is more of a “nice categories” approach than a “nice spaces” approach, because there is no known global property which would express what it means for a space to be tame. That is, there are many examples of o-minimal structures, but (it is conjectured) there is no maximal o-minimal structure, therefore no overarching meaning of what it would mean for a space to be tame.
Basically, an o-minimal structure $T$ is a collection $T_n \subseteq P(\mathbb{R}^n)$ which is closed under all first-order logic operations (e.g., complements, finite intersections, direct images under projections = existentially quantified sets, equality predicates, and the binary predicate $\lt$ on $\mathbb{R}$), and which satisfies the all-important o-minimality condition: the only sets belonging to $T_1$ are finite unions of points and intervals. The elements of $T$ may be called $T$-definable sets; the archetypal example is where $T$ is the collection of semi-algebraic sets (loci of polynomial inequalities) – cf. the Tarski-Seidenberg theorem. The thrust of the o-minimality condition is to forbid sets like $\mathbb{N} \subseteq \mathbb{R}$ from being $T$-definable, which (following Gödel, Turing, Robinson, Matiyasevich, and others) would open the door to all sorts of pathological sets being $T$-definable as well. So you could think of o-minimality as a kind of logical “monster-barring” device, which happens to be quite effective. See van den Dries’s book for a very illuminating discussion.
There are other approaches to tame topology (such as via Shiota’s $\mathcal{X}$-sets), but I am less familiar with them.
The following discussion has been acted upon by separating this page from nice category of spaces.
Tim: As I read the entry on nice topological spaces, it really refers to ‘nice categories’ rather than ‘nice spaces’! I have always thought of spaces such as CW-complexes and polyhedra as being ‘locally nice’, but the corresponding categories are certainly not ‘nice’ in the sense of nice topological space. Perhaps we need to adjust that other entry in some way.
Toby: You're right, I think I've been linking that page wrongly. (I just now did it again on homotopy type!) Perhaps we should write locally nice space? or locally nice topological space? (you pick), and I'll fix all of the links tomorrow.
Tim:I suggest locally nice space?. (For some time I worked in Shape Theory where local singularities were allowed so the spaces were not locally nice!) There would need to be an entry on locally nice. I suggets various meanings are discussed briefly, e.g. locally contractible, locally Euclidean, … and so on, but each with a minimum on it as the real stuff is in CW-complex etc and these are the ‘ideas’.
Mike: Why not change the page nice topological space to be about CW-complexes and so on, and move the existing material there to something like convenient category of spaces, which is also a historically valid term? I am probably to blame for the current misleading content of nice topological space and I’d be happy to have this changed.
Toby: I thought that nice topological space was supposed to be about special kinds of spaces, such as locally compact Hausdorff spaces, whose full subcategories of $\Sp$ are also nice. (Sort of a counterpoint to the dichotomy between nice objects and nice categories, whose theme is better fit by the example of locally Euclidean spaces). CW-complexes also apply —if you're interested in the homotopy categories.
Mike: Well, that’s not what I thought. (-: I don’t really know any type of space that is nice and whose corresponding subcategory of Top is also nice. The category of locally compact Hausdorff spaces, for instance, is not really all that nice. In fact, I can’t think of anything particularly good about it. I don’t even see any reason for it to be complete or cocomplete!
I think it would be better, and less confusing, to have separate pages for “nice spaces” and “nice categories of spaces,” or whatever we call them. And, as I said, I don’t see any need to invent a new term like “locally nice.”
When algebraic topologists (and, by extension, people talking about $\infty$-groupoids) say “nice space” they usually mean either (1) an object of some convenient category of spaces, or (2) a CW-complex-like space, between which weak homotopy equivalences are homotopy equivalences. Actually, there is a precise term for the latter sort: an m-cofibrant space, aka a space of the (non-weak) homotopy type of a CW complex.
Toby: I thought the full subcategory of locally compact Hausdorff spaces was cartesian closed? Maybe not, and it's not mentioned above.
But you can see that most of the examples above list nice properties of their full subcategories. And the page begins by talking about what a lousy category $\Top$ is. So it seems clearly wrong that you can't make $\Top$ a nicer category by taking a full subcategory of nice spaces. (Not all of the examples are subcategories, of course.)
Mike: It’s true that locally compact Hausdorff spaces are exponentiable in $Top$. However, I don’t think there’s any reason why the exponential should again be locally compact Hausdorff.
I guess you are right that one could argue that compactly generated spaces themselves are “nice,” although I think the main reason they are important is that the category of compactly generated spaces is nice. I propose the following:
Toby: I believe that the compact Hausdorff reflection (the Stone–Čech compactification) of $Y^X$ is an exponential object.
Anyway, your plan sounds fine, although nice category of spaces might be another title. (I guess that it's up to whoever gets around to writing it first.) Although I'm not sure that people really mean m-cofibrant spaces when they speak of nice topological spaces when doing homotopy theory; how do we know that they aren't referring to CW-complexes? (which is what I always assumed that I meant).
Mike: I guess nice category of spaces would fit better with the existing cumbersomely-named dichotomy between nice objects and nice categories. I should have said that when people say “nice topological space” as a means of not having to worry about weak homotopy equivalences, they might as well mean (or maybe even “should” mean) m-cofibrant space. If people do mean CW-complex for some more precise reason (such as wanting to induct up the cells), then they can say “CW complex” instead.
Re: exponentials, the Stone-Čech compactification of $Y^X$ will (as long as $Y^X$ isn’t already compact) have more points than $Y^X$; but by the isomorphism $Hom(1,Y^X)\cong Hom(X,Y)$, points of an exponential space have to be in bijection with continuous maps $X\to Y$.
Toby: OK, I'll have to check how exactly they use the category of locally compact Hausdorff spaces. (One way is to get compactly generated spaces, of course, but I thought that there was more to it than that.) But anyway, I'm happy with your plan and will help you carry it out.