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 join $X \star Y$ of topological spaces $X$ and $Y$ consists of $X$, $Y$, and convex combinations of a point of $X$ and a point of $Y$ (with some suitable topology).
Write $I \coloneqq [0,1]$ for the unit interval, regarded as a topological space.
For $X,Y$ two topological spaces, the join $X \star Y$ is the colimit in Top of the diagram
where $i_0$ is the inclusion $(x, y) \mapsto (x, 0, y)$, and $i_1$ is the inclusion $(x, y) \mapsto (x, 1, y)$.
(This in turn, by the discussion at mapping cone, is a model for the homotopy type of the homotopy pushout of the two projections $X \leftarrow X \times Y \to Y$.)
Intuitively, $X \star Y$ is the union of all line segments connecting $X$ to $Y$ when these are placed in general position in an ambient Euclidean space.
The join is associative; intuitively, a join of three spaces $X, Y, Z$ is the union of 2-simplices whose vertices lie in $X, Y, Z$ respectively when these are placed in general position. Thus the join endows $Top$ with a monoidal category structure whose unit is the empty space.
There is another topology on the join of spaces, due to John Milnor, which appears in the Milnor construction of universal principal bundles (Milnor 1956, section 2):
Milnor’s join $X \overline{\star} Y$ has the same underlying set as $X \star Y$ and the coarsest topology which makes $t \colon X \overline{\star} Y \to [0,1]$ (given on $X \times I \times Y$ by $(x,t,y)\mapsto t$, constantly $0$ on $X$, and constantly $1$ on $Y$) and the projections $\pi_1 \colon t^{-1}([0,1)) \to X$ and $\pi_2 \colon t^{-1}((0,1]) \to Y$ continuous.
If $p\colon X \to S$ and $p\colon Y \to S$ are spaces over $S$, then their fibrewise join or Whitney sum $X \overline{\star}_S Y$ is the space $\{ x \in X \overline{\star} Y : t(x) \in (0,1) \implies p(\pi_1(x))=q(\pi_2(y)) \}$.
The inclusions $X \hookrightarrow X \overline{\star} Y$ and $Y \hookrightarrow X \overline{\star} Y$ are closed Hurewicz cofibrations: the open set $t^{-1}[[0,1)]$ deforms onto $X$ and $t^{-1}[(0,1]]$ deforms onto $Y$.
The identity map $X \star Y \to X \overline{\star} Y$ is a homotopy equivalence. If $X$ and $Y$ are compact, then it is a homeomorphism.
If $p \colon X \to S$ and $q \colon Y \to S$ are Hurewicz fibrations, then so is $X \overline{\star}_S Y \to S$
The join of any topological space $X$ with the point is the cone on $X$:
The join of any topological space $X$ with the 0-sphere $S^0$ is the (unreduced) suspension of $X$:
For $X = S^1, Y = S^1$, we may consider $X$ to consist of quaternions $a + b i$ and $Y$ to consist of quaternions $c j + d k$ such that $a^2 + b^2 = 1 = c^2 + d^2$. Then $X$ and $Y$ are in general position in $\mathbb{H} \cong \mathbb{R}^4$ with respect to each other, and the quotient map
is an explicit realization of the unit sphere $S^3$ as $X \star Y$.
For a topological group $G$, the Milnor construction of the total space $E G$ of the classifying bundle is an iterated join, i.e., the colimit of a diagram of inclusions
where the identity element of $G$ is used to embed each $G^{\bar \star n}$ into its successor $G^{\bar \star (n+1)}$. The idea is that passing to higher joins kills off more and more lower-dimensional homotopy groups, until one reaches the colimit which is then weakly contractible.
The same idea applies to a general space $X$; an $H$-space structure and higher homotopy associativities (collectively embodied in a structure of algebra over the Stasheff operad) may be used to build a classifying bundle in iterative fashion, with the Hopf construction giving the first stage of the iteration.
If $p \colon X \to Y$ is a fibration, then $p$ factors as a Hurewicz cofibration $X \hookrightarrow X \overline{\star}_Y Y$ followed by an acyclic Hurewicz fibration $X \overline{\star}_Y Y \to Y$ by proposition : the embedding $Y \hookrightarrow X \overline{\star}_Y Y$ and retraction $X \overline{\star}_Y Y \to Y$ exhibit $Y$ as a strong deformation retract of $X \overline{\star}_Y Y$.
(This fact is used in Strøm’s construction of his model structure on topological spaces.)
I.M. Hall, The generalized Whitney sum, The Quarterly Journal of Mathematics 16(4): 360–384, December 1965. 10.1093/qmath/16.4.360
P.W.H. Lemmens, A note on the join of fibrations, Indagationes Mathematicae 73: 53–56 (1970) doi:10.1016/S1385-7258(70)80008-7
John Milnor, Construction of Universal Bundles, II, Ann. of Math. 63 3 (1956) 430-436 [doi:10.2307/1970012]
Discussion of relation to homotopy limits etc.
Comparison with other forms of join:
Discussion in homotopy type theory (applied to n-image factorization) is in
Last revised on April 30, 2023 at 09:18:59. See the history of this page for a list of all contributions to it.