see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
Basic homotopy theory
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
The free loop space $\mathcal{L}X$ of a topological space $X$ (based or not) is the space of all loops in $X$. This is in contrast to the based loop space of a based space $X$ for which the loops are at the fixed base point $x_0\in X$.
The free loop space carries a canonical action (infinity-action) of the circle group. The homotopy quotient by that action $\mathcal{L}(X)/S^1$ (the “cyclic loop space”) contains what is known as the twisted loop space of $X$.
For $X$ a topological space, the free loop space $L X$ is the topological space $Map(S^1,X)$ of continuous maps in compact-open topology.
If we work in a category of based spaces, then still the topological space $Map(S^1,X)$ is in the non-based sense but has a distinguished point which is the constant map $t\mapsto x_0$ where $x_0$ is the base point of $X$.
If $X$ is a topological space, the free loop space $L X$ of $X$ is defined as the free loop space object of $X$ formed in the (∞,1)-category Top.
Let $X$ be a simply connected topological space.
The ordinary cohomology $H^\bullet$ of its free loop space is the Hochschild homology $HH_\bullet$ of its singular chains $C^\bullet(X)$:
Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:
(Loday 11)
If the coefficients are rational, and $X$ is of finite type then this may be computed by the Sullivan model for free loop spaces, see there the section on Relation to Hochschild homology.
In the special case that the topological space $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that
This is known as Jones' theorem (Jones 87)
An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.
See at Sullivan model of free loop space.
loop space object, free loop space object,
loop space, free loop space, derived loop space
Kathryn Hess, Free loop spaces in topology and physics, pdf, slides from Meeting of Edinburgh Math. Soc. Glasgow, 14 Nov 2008
John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf)
Jean-Louis Loday, Free loop space and homology (arXiv:1110.0405)
D. Ben-Zvi, D. Nadler, Loop spaces and connections, arxiv/1002.3636