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
An open cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of open subsets of $X$ whose union equals $X$: $\cup_i U_i = X$.
When denoting by $U_i \hookrightarrow X$ the inclusion morphisms in the category Top, each open cover constitutes a covering family $\{U_i \to X\}$ in the sense of sheaf and topos theory which is a standard coverage on Top.
Analogous statements hold for categories of topological spaces with extra structure, such as the category Diff of smooth manifolds.
If an open cover has the property that all the $U_i$ and all of their finite nonempty intersections are contractible, then one speaks of a good open cover.
Last revised on November 11, 2021 at 07:46:08. See the history of this page for a list of all contributions to it.