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 functional analysis and related areas of mathematics stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. They form a class of spaces with a series of remarkable properties, in particular, this class is very wide (for instance, it contains all Fréchet spaces and thus, all Banach spaces), it consists of spaces satisfying a natural condition of completeness, and it forms a closed monoidal category with the standard analytical tools for constructing new spaces, like taking closed subspace, quotient space, projective and injective limits, the space of operators, tensor products, etc.
Let $X$ be a topological vector space over the complex numbers $\mathbb{C}$. Let $X^\ast$ be the vector space of continuous linear functionals $X \to \mathbb{C}$ endowed with the topology of uniform convergence on totally bounded subsets of $X$^{1}. This $X^\ast$ is a topological vector space, and may be regarded as a form of dual space of $X$. This construction defines a contravariant functor $(-)^\ast$ on the category of topological vector spaces.
There is a canonical natural transformation $i_X: X \to X^{\ast\ast}$, sending $x \in X$ to the functional defined by $i(x)(f) \coloneqq f(x)$.
A TVS $X$ is a stereotype space if $i_X$ is an isomorphism.
There is another way of characterizing stereotype spaces $X$ directly in terms of the TVS structure on $X$. We say that a TVS $X$ is pseudo-complete if every totally bounded Cauchy net in $X$ converges. Next, call a subset $C$ of $X$ capacious if, for every totally bounded set $A \subset X$, there exists a finite set of elements $x_1, \ldots, x_n$ such that every $y \in \bar{A}$ is contained in some translate $x_i + C$. Then, say that $X$ is pseudo-saturated if every closed convex balanced capacious subset of $0$ is a neighborhood of $0$ (this can be regarded as a weakening of the notion of barreledness).
A TVS $X$ is a stereotype space if and only if it is locally convex, pseudo-complete, and pseudo-saturated.
The inclusion of pseudo-saturated locally convex TVS, as a full subcategory of locally convex TVS, is coreflective. The right adjoint to the inclusion takes $X$ to a space denoted $X^\triangle$, called the pseudo-saturation of $X$. If $X$ is pseudo-complete, so is $X^\triangle$.
Let $X$, $Y$ be stereotype spaces, and let $L(X, Y)$ be the space of continuous linear maps $X \to Y$. Then $L(X, Y)$ is locally convex, and as $Y$ is pseudo-complete, it follows that $L(X, Y)$ is also pseudo-complete. However, $L(X, Y)$ might not be pseudo-saturated. Let $Hom(X, Y)$ be the pseudo-saturation of $L(X, Y)$, so that $Hom(X, Y)$ is a stereotype space. This construction endows the category of stereotype spaces with a closed category structure, with unit $\mathbb{C}$. We have $X^\ast \cong Hom(X, \mathbb{C})$.
The category of stereotype spaces, as a closed category with duality $(-)^\ast$, acquires a structure of complete and cocomplete $\ast$-autonomous category.
In slightly more detail, a tensor product on stereotype spaces may be defined by the formula $X \otimes Y \coloneqq Hom(X, Y^\ast)^\ast$. Then for stereotype spaces $X, Y, Z$ there is a natural isomorphism
and there are associativity, symmetry, and unit constraints on $\otimes$ so that in this way, stereotype spaces form a symmetric monoidal closed category. By taking $D = I = \mathbb{C}$ as dualizing object, it is moreover $\ast$-autonomous. It is not, however, autonomous in the sense of compact closed categories, i.e., we do not have an isomorphism between $(X \otimes Y)^\ast$ and $X^\ast \otimes Y^\ast$. Rather, we get a second tensor product via the formula
and this tensor product plays the role of multiplicative disjunction in linear logic (where we regard $\ast$-autonomous categories as models of multiplicative linear logic).
In other words, the smallest topology on $X^\ast$ that includes all sets $L(B, U) \coloneqq \{f: X \to \mathbb{C}: f(B) \subseteq U\}$ where $B$ is totally bounded in $X$ and $U$ is open in $\mathbb{C}$. ↩
Last revised on November 10, 2013 at 11:08:00. See the history of this page for a list of all contributions to it.