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 category TopSp of all topological spaces serves in the foundations of many subfields of mathematics, as reflected in the widely (but not universally) accepted convention that “space” in mathematics means “topological space”, by default. However, the great flexibility of the notion of topological spaces also has its drawbacks, as without further qualification it tends to be a little too general for usage in any given sub-field. At the same time, the relevant further conditions typically required on topological spaces in any given application varies, sometimes considerably, and many further such conditions have been and are being introduced. Hence what counts as a well-behaved (“nice”) topological spaces strongly depends on context, and the main point to be aware of is the existence of key classes of possible extra conditions on topological spaces.
The foremost among these are certainly the separation axioms, which may be understood as prescribing how exotic (pathological) a topological space is allowed to be, locally, as compared to the archetypical example of Euclidean space (which, it is worthwhile to remember, was the default meaning of “space” for most of human history). In consequence, separation conditions are typically required of topological spaces if and to the extent that these are used as models for space in the original sense of geometry. Probably the most prominent among the separation axioms is the Hausdorff-condition “$T_2$”, and many authors will by default mean “Hausdorff topological space” when saying “topological space”.
Another class of nicety conditions on topological spaces are not primarily motivated by the look of the individual spaces, but by ensuring that the (preferably (co-)reflective) full subcategory of spaces satisfying these conditions has good category theoretic properties, such as cartesian closure (but beware that there is a dichotomy between nice objects and nice categories). Such convenient categories of topological spaces are of foundational importance in algebraic topology where it is tolerable to change the homeomorphism-type of a space as long as its homotopy type is retained. Prominent examples of classes of topological spaces which are “nice” in a way that their categories are “convenient”, in this sense, are the “subcategory-generated spaces”, namely those that may be obtained as colimits of building block spaces taken from a given full subcategory. For example, if these building blocks are taken to be all compact spaces then one speaks of compactly generated topological spaces, while if they are taken to be all Euclidean spaces one speaks of Delta-generated topological spaces.
Further in this vein of algebraic topology and homotopy theory are conditions on a topological space which ensure that it is a good model for its (weak) homotopy type. For example, the condition that a topological space be a CW-complex (hence cofibrant in the classical model structure) ensures that continuous functions out of it model all homotopy classes of morphisms out of the homotopy type (as witnessed by Whitehead's theorem), while the condition that a topological group be well-pointed (hence cofibrant in the pointed Strøm model category) ensures that the topological realization of its nerve has the expected homotopy type. It is this last condition which leads over to the notion of nice simplicial topological spaces.
The following is a list – currently somewhat random and highly incomplete – of nicety conditions that are considered on topological spaces.
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 locally Euclidean space such as 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.
Discussion of notorious subtleties: