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 topological space is called irreducible if it cannot be expressed as a finite union of proper closed subsets, or equivalently if any finite collection of inhabited open subsets has inhabited intersection.
If is irreducible according to Def. , then:
cannot be expressed as the union of two proper closed subsets. Equivalently, any two inhabited open subsets have inhabited intersection.
cannot be expressed as the union of the empty collection of proper closed subsets, which implies that must be inhabited. Equivalently, the intersection of the empty collection of inhabited open subsets has inhabited intersection, which means exactly that is inhabited.
(role of the empty set)
Another common version of Def. (cf. Wikipedia) is to have it speak just about binary unions, as opposed to finite unions. With that definition the empty set would be irreducible, while according to Def. it is not: The empty set is the empty union and hence a finite union of proper closed subsets, but not a binary union of such, since it has no proper subsets!
But having the empty set count as reducible is useful, cf. Ex. below.
A subset of a topological space is an irreducible subset if is an irreducible topological space with the subspace topology.
An algebraic variety is irreducible if its underlying topological space (in the Zariski topology) is irreducible.
A sober topological space, is one whose only irreducible closed subsets are the closures of single points.
See also:
Last revised on June 5, 2025 at 15:22:55. See the history of this page for a list of all contributions to it.