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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A pointed topological pair is a pointed topological space $X=(X,x_0)\in Top_*$ equipped with a topological subspace $A\subset X$ containing the base point $x_0$.
Let $I=[0,1]$ be the unit closed interval and for $N\in\mathbb{N}$ let $J^N \coloneqq \partial I^N-\{0\}\times I^{N-1}$.
The relative loop space of a pointed topological pair $(X,A)$ is the space of continuous maps of the form $(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0)$, denoted $\Omega^N(X,A)$.
For $A = \{x_0\}$ this reduces to the ordinary notion of (iterated) loop space.
The relative loop spaces allow us to define the relative homotopy groups of topological pairs.
For $N\geq 1$ the $N$-th relative homotopy set of a topological pair $(X,A)$, denoted $\pi_N(X,A)$ is defined as the set of connected components of the associated relative loop space, ie $\pi_N(X,A)\coloneqq \pi_0\Omega^N(X,A)$.
If $N\geq 2$ then $\pi_N(X,A)$ is a group, which is abelian if $N\geq 3$.
If we are interested in the homotopy theory of relative loop spaces (as in recognition of relative loop spaces) the above definition is not appropriate since there is no model category-structure on the category of topological pairs, as explained in this mathoverflow discussion.
The solution here is to work in the category $Top^\to_*$ of continuous pointed maps equipped with the projective model structure on functors. If we start with the Quillen model structure on $Top$, the cofibrant objects in $Top^\to_*$ are the inclusions of CW-pairs, and if we start with the mixed model structure we get the maps homotopy equivalent to those.
We can then define relative loop spaces as loop spaces of homotopy fibers.
For $N\geq 1$ the relative $N$-loop space functor is the right derivable functor
For inclusions of topological pairs the two definitions of relative loop spaces are naturally homeomorphic.
Last revised on March 29, 2023 at 01:41:49. See the history of this page for a list of all contributions to it.