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
Given a set $\{X_i\}_{i \in I}$ of topological spaces, then the box topology on the Cartesian product $\underset{i \in I}{\prod} X_i$ of the underlying sets of these spaces is the topology which is generated from the topological base whose elements are the Cartesian products
of open subsets $U_i \subset X_i$ of each of the factor spaces.
If the index set $I$ is a finite set, then this box topology coincides with the Tychonoff topology on the Cartesian product. For general $I$ however the Tychnoff topology has as base open subsets only those products $\underset{i \in I}{\prod} U_i$ for which all but a finite number of factors are in fact the corresponding total space $X_i$.
Hence for non-finite index set $I$ the box topology is strictly finer that the Tychonoff topology. Beware that it is the Tychonoff toopology which yields the actual Cartesian product in the category Top of topological spaces. Accordingly, for non-finite $I$ the box topology fails to possess the properties that one expects from a categorical product.
For example, there may be a family of continuous functions, $f_i: X \to X_i$, for which the associated map by components, $X \to \underset{i\in I}{\prod} X_i$, is not continuous for the box topology.
Last revised on May 4, 2017 at 12:31:54. See the history of this page for a list of all contributions to it.