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
Bohr compactification is a kind of compactification of a topological group to one that is compact Hausdorff.
Let $i: CHGrp \to TopGrp$ denote the full embedding of the category of compact Hausdorff topological groups into the category of all topological groups. Let $G$ be a topological group.
The Bohr compactification $Bohr(G)$ is a representing object for the functor $TopGrp(G, i-): CHGrp \to Set$.
The Bohr compactification exists for any topological group $G$.
We verify that the hypotheses of the general adjoint functor theorem are satisfied. Certainly $CHGrp$ is complete, and $i: CHGrp \to TopGrp$ preserves small limits since products and equalizers in $CHGrp$ are formed just as they are in $TopGrp$. We must now show there is a small solution set, i.e., a weakly initial set for the comma category $G \downarrow i$.
Let $I = [0, 1]$ be the unit interval with its standard topology. There is a canonical map $u_G: G \to I^{Top(G, I)}$ where $u: id \to M$ is the unit of the monad $M = I^{Top(-, I)}$ on $Top$; note $u_X$ is a closed embedding if $X$ is compact Hausdorff (see the discussion here). We claim that the collection of closed subspaces of $I^{Top(G, I)}$ that come equipped with topological group structures is a solution set. Indeed, for an arbitrary CH group $K$ and continuous homomorphism $f: G \to K$, the closure of the image $\widebar{f(G)}$ is a compact Hausdorff subgroup of $K$ through which $f$ factors (according to the lemma that follows), and this is a closed subspace of $M(f)^{-1}(u_K)$ in the pullback diagram
and therefore $\widebar{f(G)}$ occurs as a closed subspace of $I^{Top(G, I)}$, as claimed.
If $H$ is a topological group and $J \subseteq H$ is a subgroup, then the topological closure $\widebar{J} \subseteq H$ is also a subgroup. (The same is true for any finitary algebraic theory in place of the theory of groups.)
For $X, Y$ arbitrary topological spaces and subsets $A \subseteq X$, $B \subseteq Y$, it is elementary that $\widebar{A \times B} = \widebar{A} \times \widebar{B}$. Hence for any operation $m: H^n \to H$ that $J$ is closed under, $\widebar{J}^n = \widebar{J^n}$ is contained in $m^{-1}(\widebar{J})$ (by continuity of $m$), so that $m$ restricts to an operation $\widebar{J}^n \to \widebar{J}$ as required.
The Bohr compactification admits a more elegant construction if $G$ is a topological abelian group: if $S = S^1$ is the unit circle, then $Bohr(G)$ may be taken to be the closure of the image of $G$ under the canonical map $G \to S^{TopAb(G, S)}$. The argument is that it suffices to consider only compact Hausdorff abelian groups $K$, where $K$ embeds as a closed subgroup of $S^{TopAb(K, S)}$ by Pontryagin duality; using a pullback similar to the above, the argument is easily completed. The description simplifies further if $G$ is a locally compact Hausdorff abelian group; here $Bohr(G)$ is the Pontryagin dual of the discretization of the Pontryagin dual of $G$.
A MathOverflow question from 2011 asks whether, for $G$ a compact Hausdorff group, can $\mathbb{Z}$ appear as a quotient of $G$ considered as an abstract group?
A truly simple answer was given by Sean Eberhard using the Bohr compactification, as follows. A quotient $p: G \to \mathbb{Z}$ admits a section $i: \mathbb{Z} \to G$ which extends to the Bohr compactification $\widehat{i}: Bohr(\mathbb{Z}) \to G$. The composite $p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z}$ is still surjective as its restriction along the unit $\mathbb{Z} \to Bohr(\mathbb{Z})$ is the identity. According to the remark above, $Bohr(\mathbb{Z})$ is the Pontryagin dual of the discrete group $S^1 \cong \mathbb{R}_{disc} \oplus \mathbb{Q}/\mathbb{Z}$, so $Bohr(\mathbb{Z}) \cong \mathbb{R}_{disc}' \oplus (\mathbb{Q}/\mathbb{Z})'$. Now the Pontryagin dual of a discrete torsionfree group such as $\mathbb{R}_{disc}$ is divisible (more generally, if $Q$ is an injective module over a commutative ring and $F$ is flat, then $\hom(F, Q)$ is also injective), so the restriction of $p \widehat{i}: Bohr(\mathbb{Z}) \to \mathbb{Z}$ to the summand $\mathbb{R}_{disc}'$ is zero. But the restriction to the other summand $(\mathbb{Q}/\mathbb{Z})' \cong \widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p^\wedge$ is also zero, as the further restriction to $\mathbb{Z}_2^\wedge$ is zero by 3-divisibility, and the restriction to $\prod_{p \neq 2} \mathbb{Z}_p^\wedge$ is zero by 2-divisibility.
Last revised on August 23, 2018 at 01:24:48. See the history of this page for a list of all contributions to it.