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 Jordan curve theorem is a basic fact in topology. It stands out as having a statement that intuitively seems completely obvious, but which turns out to require a non-trivial formal proof.
The theorem states that every continuous loop (where a loop is a closed curve) in the Euclidean plane which does not intersect itself (a Jordan curve) divides the plane into two disjoint regions?, the inside contained in the curve, and the outside.
Last revised on September 6, 2017 at 07:19:12. See the history of this page for a list of all contributions to it.