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
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 quotient space
of the product space $X \times I \times Y$ by the equivalence relation
In other words, the join 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.
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 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 $G^{\ast n}$ into its successor $G^{\ast (n+1)}$. 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.
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 May 18, 2020 at 16:45:58. See the history of this page for a list of all contributions to it.