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
A $\sigma$-topological space is a set $X$ equipped with a $\sigma$-topology: a collection $O(X)$ of subsets of $X$, called open sets, which is closed under finite intersections and countable unions.
The open sets $O(X)$ of a $\sigma$-topological space form a $\sigma$-frame.
An example of a $\sigma$-topological space is a $\sigma$-algebra, whose collection $O(X)$ is also closed under complements and countable intersections. Another example of a $\sigma$-topological space is a zero-set structure.
Let $X = \prod_{n \to \alpha} X_n$ be an internal set, and let $\mathcal{F}_X$ be the collection of all subsets of $X$ that can be expressed as the union of at most countably many internal subsets of $X$. Then $(X, \mathcal{F}_X)$ is a countably compact $T_1$ $\sigma$-topological space.
Fedor Petrov, “countable” topology, MathOverflow (web)
Vitaly Bergelson?, Terence Tao, Multiple recurrence in quasirandom groups. (arXiv:1211.6372)
Last revised on February 4, 2024 at 18:33:45. See the history of this page for a list of all contributions to it.