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
In topology, a separable metric space is a topological space that is both separable and metrizable.
For separable metric spaces, the following notions of dimension all (exist and) coincide and are thus uniformly referred to as the dimension of a separable metric space:
small and large inductive dimension;
(e.g. Engelking 78, Theorem 1.7.7)
Ryszard Engelking, Dimension Theory, Mathematical Library 19, North-Holland Publishing/Polish Scientific Publishers 1978 (pdf)
Ryszard Engelking, Theory of Dimensions – Finite and Infinite, Sigma Series in Pure Mathematics 10, Helderman 1995 (pdf)
Andrej Bauer and Andrew Swan, Every metric space is separable in function realizability, 2018, (arxiv:1804.00427)
Last revised on August 29, 2024 at 20:06:16. See the history of this page for a list of all contributions to it.