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 convergence vector space is a generalisation of a topological vector space where the topological structure is replaced by a convergence structure.
A convergence vector space is a vector space, say $V$, together with the structure of a convergence space, say $\Lambda$, which is compatible with the vector space structure in that both addition and scalar multiplication are jointly continuous.
Some authors may impose the condition that the convergence structure be separated. Any topological vector space defines a convergence vector space.
Convergence structures become particularly useful when considering evaluation mappings in functional analysis. For a locally convex topological vector space $E$ with continuous dual $E^*$, the evaluation mapping $E \times E^* \to \mathbb{F}$ is continuous with respect to the product topology if and only if $E$ is normable. This proves to be somewhat of a limitation when one works outside the category of normable LCTVS. However, there is a convergence structure on $E \times E^*$ which makes the evaluation mapping continuous.
Let $E$ and $F$ be convergence vector spaces and $n \in \mathbb{N}$. Let $\mathcal{L}^n(E,F)$ be the space of continuous mappings $E^n \to F$ which are linear in each argument (see continuous multilinear operator). The continuous convergence structure on $\mathcal{L}^n(E,F)$ is the coarsest convergence structure which makes the evaluation map $\mathcal{L}^n(E,F) \times E^n \to F$ continuous.
We write $\mathcal{L}_c^n(E,F)$ for the resulting convergence space.
The convergence space $\mathcal{L}_c^n(E,F)$ is a convergence vector space. It satisfies the following universal property for a convergence space $X$ then a mapping $X \to \mathcal{L}_c^n(E,F)$ is continuous if and only if the associated map $X \times E^n \to F$ is continuous.
Last revised on May 23, 2013 at 14:37:28. See the history of this page for a list of all contributions to it.