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
For $n \in \mathbb{N}$ a natural number, write $\mathbb{R}^n$ for the Cartesian space of dimension $n$. The Euclidean topology is the topology on $\mathbb{R}^n$ characterized by the following equivalent statements
it is the metric topology induced from the canonical structure of a metric space on $\mathbb{R}^n$ with distance function given by $d(x,y) = \sqrt{\sum_{i = 1}^n (x_i-y_i)^2}$;
an open subset is precisely a subset such that contains an open ball around each of its points;
it is the product topology induced from the standard topology on the real line.
Two Cartesian spaces $\mathbb{R}^k$ and $\mathbb{R}^l$ (with the Euclidean topology) are homeomorphic precisely if $k = l$.
A proof of this statement was an early success of algebraic topology.
Last revised on June 10, 2021 at 22:43:20. See the history of this page for a list of all contributions to it.