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 dyadic space [Engelking 1989, p. 231] is a compact topological space which is homeomorphic to the image under a continuous map of a Cantor cube , for some infinite set.
A dyadic compactum [Shapiro 2003] is a dyadic space which is also Hausdorff (hence a dyadic compact Hausdorff space and in this sense a “compactum”).
Ryszard Engelking, pp. 231, 232, 291 in: General Topology, Sigma Series in Pure Mathematics 6, Heldermann (1989) [ISBN:388538-006-4, pdf]
Leonid B. Shapiro: Dyadic Compacta, Encyclopedia of General Topology (2003) 192-194 [doi:10.1016/B978-044450355-8/50052-0]
See also:
Wikipedia: Dyadic space
Last revised on August 18, 2025 at 09:13:34. See the history of this page for a list of all contributions to it.