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
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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
In topology the path space of a topological space $X$ is a topological space of all continuous paths in $X$.
In topological homotopy theory the path space constrution serves to exhibit homotopies in the guise of right homotopies. This situation generalizes to many other model categories and one speaks more generally of path space objects in this case.
For exposition in the context of point-set topology see at Introduction to Topology -- 1 around this example.
For exposition in the context of topological homotopy theory see at Introduction to Homotopy Theory around this definition.
For $X$ a topological space, then its path space is the mapping space $X^{[0,1]}$, out of the topological interval into $X$, i.e. the set of continuous function $\gamma \;\colon\; [0,1] \to X$ equipped with the compact-open topology.
The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions
(inclusion of constant paths and endpoint evaluation of paths).
The fiber product of the projection with the diagonal on $X$ is the free loop space $\mathcal{L X}$ of $X$:
If $X$ is equipped with a choice of basepoint $x \colon \ast \to X$ (making it a pointed topological space), then the further fiber product with this basepoint inclusion is the based loop space $\Omega_x X$:
See also