see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
Basic homotopy theory
The profinite completion $\hat{G}$ of a (discrete) group $G$ is the limit (in the category of topological groups) over the diagram with objects all the finite quotient groups $G/N_{fin}$ where $N_{fin}$ is a normal subgroup of $G$ with finite index, and morphisms induced from the lattice of subgroups of $G$.
Note that the profinite completion actually is a profinite group, and there is a canonical homomorphism $G \to \hat{G}$.
More formally, we note that for any group $G$, the family of its normal finite index subgroups forms a cofiltered category under inclusion. (Denote it by $\Omega_G$.) The assignment of $G/N$ to $N$ gives a functor from $\Omega_G^{op}$ to the category of finite groups. It is thus a profinite group in the sense given in that entry, i.e. a pro-object in the category of finite groups. This pro-object is the profinite completion of $G$.
The above topological version of this, with which we started, is obtained by means of the equivalence between the category of pro-(finite groups) and that of the groups internal to profinite spaces that is by taking the limit in the category of topological groups of the diagram of (discrete) finite groups that the above construction gives one.
The inclusion functor $inc$ from the category, $FinGrps$, of finite groups, into that of groups does not have a left adjoint. It does have a left pro-adjoint, that is to say, it induces a functor on procategories
and that functor does have a left adjoint. If we restrict that ‘pro-adjoint’ to the subcategory of $pro\!-\! FinGrps$ given by the ‘constant’ pro-objects, then the result is the pro-finite completion construction that is given above. Because of this, if we think of the natural functor $C\to pro\!-\! C$ to be an inclusion, i.e. think of an object as a pro-object indexed by the one arrow category, we can give a universal property for the pro-finite completion of a group $G$. This universal property gives a universal cone from $G$ to finite groups, and just encodes the obvious fact that any homomorphism from $G$ to a finite group factors through one of its finite quotient groups. If we write $\hat{G}$ for the pro-finite completion, the universal cone is a map $G\to \hat{G}$ in $pro\!-\! Grps$.
Let $\mathcal{C}$ be any class of finite groups that is closed under the formation of subgroups, homomorphic images and group extensions.
A pro-$\mathcal{C}$ group is an inverse limit of an inverse system of groups in the class $\mathcal{C}$ or alternatively a pro-object in the full subcategory $\mathcal{C}$ determined by the class $\mathcal{C}$.
The subcategory of $pro\!-\!FinGrps$ consisting of the pro-$\mathcal{C}$ groups and the continuous homomorphisms between them will be denoted $pro\!-\!\mathcal{C}$.
This notation now has two definitions, but, as the corresponding categories are equivalent, this causes no problem.
The categories of the form $pro\!-\!\mathcal{C}$ form varieties in $prof-FinGrps$. Recall that a variety in any algebraic context means a subcategory of ‘algebras’ closed under products, subobjects and quotients. We note the condition on $\mathcal{C}$ implies the closure of $\mathcal{C}$ under finite products, so $\mathcal{C}$ is what is called a pseudovariety. The category $pro\!-\!FinGrps$ is monadic over the category of spaces. This means that free objects exist in all the $pro\!-\!\mathcal{C}$. A good reference for this is Gildenhuys and Kennison, (1971), see below.
Consider the profinite completion of the fundamental group of an complex projective variety $X$. Since $X$ has an underlying topological space, its fundamental group of loops $\pi_1^{top}(X)$ can be defined in the usual way. But one can also define the algebraic fundamental group $\pi_1^{alg}(X)$. This is a profinite group, which is isomorphic to the profinite completion of $\pi_1^{top}(X)$.
The profinite completion of the integers is
This is isomorphic to the product of the p-adic integers for all $p$
For more on this see at p-adic integers, at adele and idele.
(Beware there are two possible interpretations of this term. One is handled in the section above, being profinite completion of the homotopy type of a space. The entry linked to here treats another more purely topological concept.)
Profinite completion of groups is a special case of a general process that ‘completes’ a category together with a ‘forgetful functor’ to some ‘base’ category, replacing it by a category which is equational/monadic over the base.
compact Hausdorff rings are profinite
Artin and Mazur in their lecture note on étale homotopy introduced a process of profinite completion, generalising that for groups in as much as the profinite completion of an Eilenberg-Mac Lane space having $G$ as fundamental group has the profinite completion of $G$ as its fundamental group. (WARNING: This needs a bit more detail to make it true! so this part of the entry needs more work.)
L. Ribes and P. Zalesskii, 2000, Profinite groups , volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge , Springer-Verlag, Berlin.
J. Dixon, M. du Sautoy, A. Mann and D. Segal, 1999, Analytic pro-p groups, volume 61 of Cambridge Studies in Advanced Mathematics , Cambridge Univ. Press.