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
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A topological ∞-groupoid is meant to be an ∞-groupoid that is equipped with topological structure. For instance a 0-truncated topological $\infty$-groupoid should just be a topological space, and 1-truncated topological $\infty$-groupoids should reproduce topological groupoids/topological stacks, etc.
A generic way to make sense of this in a gros topos perspective is to pick some small subcategory $Top_{sm}$ of the category Top of topological spaces, regard it as a site with respect to an evident coverage by open covers, and then take the (∞,1)-topos of generalized topological $\infty$-groupoids to be the (∞,1)-category of (∞,1)-sheaves on this site:
This gives a nice category with very general objects. In there one may find smaller, less nice categories of nicer objects.
There are different choices of sites $Top_{sm}$ to make. For instance
taking $Top_{sm}$ to be a small site of locally contractible topological spaces yields a concept of locally contractible topological infinity-groupoids;
taking $Top_{sm}$ be the the subcategory of topological Cartesian spaces yield a concept of Euclidean-topological infinity-groupoids.
These two hapenns to constitute cohesive ∞-toposes, due to the local contractibility of the objects in the site.