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 Sorgenfrey line (after Robert Sorgenfrey?, a mathematical descendant of R.L. Moore?) is the real line , but topologized by taking as topological basis the collection of half-open intervals . This is sometimes called the lower limit topology?, and the Sorgenfrey line is often denoted by .
The Sorgenfrey line and spaces derived from it, notably the Sorgenfrey plane (with the product topology), is a rich source for counterexamples in general topology. For example, the Sorgenfrey line is a () normal space and even a paracompact space, but the Sorgenfrey plane fails to be normal (and thus is not paracompact either). The Sorgenfrey plane is also an example of a separable space, but admitting an uncountable discrete subspace (which is thus not separable), showing that separability is not a hereditary property.
Last revised on May 22, 2017 at 19:20:01. See the history of this page for a list of all contributions to it.