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 scattered space is a topological space that on the scale of ‘connectivity’ is at the extreme opposite of perfect spaces.
Scattered spaces are used to provide topological models of provability logic.
A subset $A$ of a topological space is called dense in itself if each point of $A$ is a limit point of $A$.
A topological space $X$ is called scattered if $X$ doesn’t contain nonempty dense-in-itself subsets.
The Sierpinski space $X=\{a,b\}$ with topology $\{\emptyset ,\{a\}, X\}$ is scattered: whereas $b$ is a limit point of $X$, $a$ isn’t, whence $X$ fails to be dense in itself. $\{a\}$ fails to be dense in itself since, whereas $b$ is a limit point, $a$ isn’t. Finally, $\{b\}$ has no limit points at all.
Discrete spaces are scattered.
Given a space $X$, using the derivative $\partial : 2^X\to 2^X$ that maps a subset to the set of its limit points, the property of being dense-in-itself amounts to $A\subseteq\partial A$. Sierpiński (1927) studies the abstract properties of dense-in-itself and corresponding scattered subsets relative to arbitrary monotone maps $2^X\to 2^X$.
L. Beklemishev, D. Gabelaia, Topological interpretations of provability logic , arXiv:1210.7317 (2012). (abstract)
W. Sierpiński, La notion de derivée comme base d’une théorie des ensembles abstraits , Math. Ann. 97 (1927) pp.321-337. (gdz)
S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, p.219)