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
The Goldman bracket of a compact closed surface $\Sigma$ is a Lie algebra structure on the free abelian group generated from the isotopy classes of based loops in $\Sigma$.
Equivalently, the Goldman bracket on $\Sigma$ is a structure on the 0th homology $H_0(L \Sigma)$ of the free loop space of $\Sigma$. It is in fact just the lowest degree of the string topology operations on $\Sigma$. See there for more details.
Let $\Sigma$ be a compact closed and oriented surface (manifold of dimension 2). For $\gamma : S^1 \to \Sigma$ a continuous function from the based circle, write $[\gamma]$ for the corresponding isotopy class.
For $[\gamma_1]$ and $[\gamma_2]$ two such classes, one can always find differentiable representatives $\gamma_1$ and $\gamma_2$ that intersect - if they intersect at some point $p$ - transversally. Write $\gamma_1 \ast_p \gamma_2$ for the curve obtained by starting at the intersection point $p$, traversing along $\gamma_1$ back to that point and then along $\gamma_2$.
The Goldman bracket on the free abelian group on classes $[\gamma]$ is defined by
where $sgn(p)$ is +1 if $T_p \gamma_1, T_p \gamma_2$ is an oriented basis of the tangent space $T_p \Sigma$, and -1 otherwise.
The original definition is due to
The relation to string topology is due to