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 function $f$ is continuous if, roughly speaking, $f(x)$ is arbitrarily close to $f(y)$ whenever $x$ is sufficiently close to $y$. However, ‘close’ is relative, and $f(x)$ may be much closer to $f(y)$ than $g(x)$ is to $g(y)$, even if both $f$ and $g$ are continuous. Nevertheless, given a family of functions, we may have that $f(x)$ is arbitrarily close to $f(y)$ for every function $f$ in the family at once whenever $x$ is sufficiently close to $y$. In this case, the family of functions is equicontinuous.
Because we are considering the relative degree of closeness between potentially unrelated pairs of points, we need a uniform structure to define this concept. So let $X$ and $Y$ be uniform spaces (although the concept should make sense in somewhat greater generality), and let $\mathcal{F}$ be a family of functions from $X$ to $Y$.
The family $\mathcal{F}$ is continuous if each member is continuous: For each entourage $E$ in $Y$, for each function $f$ in $\mathcal{F}$ and each point $x \in X$, for some entourage $D$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
In short:
The family $\mathcal{F}$ is uniformly continuous if each member is uniformly continuous: For each entourage $E$ in $Y$, for each function $f$ in $\mathcal{F}$, for some entourage $D$ in $X$, for each point $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
In short:
The family $\mathcal{F}$ is equicontinuous if: For each entourage $E$ in $Y$, for each point $x$ in $X$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
In short:
The family $\mathcal{F}$ is uniformly equicontinuous if: For each entourage $E$ in $Y$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$ and each point $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
In short:
All of these definitions are identical except for the placement of the quantifiers $\forall f$ and $\forall x$ before or after the quantifier $\exists D$.
Just as it is reasonable to speak of a single function $f$ continuous at a single point $x$, so it is reasonable to speak of a family equicontinuous at a single point $x$ (or, for that matter, of a single function $f$ uniformly continuous). However, it makes no sense to speak of a single function $f$ equicontinuous (or uniformly equicontinuous), nor can we speak of a family of functions uniformly equicontinuous at a single point $x$.
The Cauchy sum theorem holds for an equicontinuous family of functions (from the real line to itself), without the requirement for uniform convergence.
The importance of equicontinuity is perhaps best illustrated by the Arzelà-Ascoli theorem, which gives conditions for a set in a function space to be compact. A reasonably general version is this:
Let $X$ be a convergence space and $Y$ a uniform space. Then a subset $F \subseteq C(X, Y)$ is relatively compact (has compact closure) iff it is equicontinuous and $\{f(x):\; f \in F\}$ is relatively compact in $Y$ for each $x \in X$.
See BB, corollary 2.4.9. Here the topology on the space $C(X, Y)$ of continuous functions $X \to Y$ is the so-called natural topology, namely the largest topology on $C(X, Y)$ such that for all spaces $A$, the continuity of a map $g: A \times X \to Y$ implies the continuity of its transpose $\hat{g}: A \to C(X, Y)$. This is the same as the exponential in $Top$ whenever the exponential exists. See Escardó, sections 8.1 and 10.2; see also ELS where it is shown that the natural topology on $C(X, Y)$ coincides with the topology of continuous convergence (which is the context for the theorem above).
(Some applications of Arzela-Ascoli should also be given.)
It happens that the property of a family of functions being (uniformly) equicontinuous is equivalent to them defining a single function which is (uniformly) continuous.
Recall that for a family of functions $f_i : A \to B$ for $i \in I$, we can define a function $\hat f : A \to B^I$ by $\hat f(a)(i) = f_i(a)$, and that the $f_i$ are all (uniformly) continuous iff $\hat f$ is (uniformly) continuous when $B^I$ is given the product uniformity.
However, we can give $B^I$ another uniformity more analogous to the box topology of topological spaces. Specifically, the box power $B^{\square I}$ is the uniform space with point set $B^I$, and where $E \subseteq B^I \times B^I$ is an entourage iff there is some entourage $G$ on $B$ such that for all $f \sim_E g$ in $B^I$ and all $i \in I$, $f(i) \sim_G g(i)$. This means that the entoruages of $B^{\square I}$ are generated by $G^I$ for entourages $G$ on $B$.
With this definition, we can see that the family of functions $f_i : A \to B$ is (uniformly) equicontinuous iff $\hat f : A \to B^{\square I}$ is (uniformly) continuous.
This also gives a convenient way to show that an equicontinuous net of continuous functions which converge pointwise converges to a continuous function: a net of equicontinuous functions $f_d : A \to B$ gives a continuous function $A \to B^{\square D}$. By the below proposition, if the $f_d$ converge pointwise, they converge to a continuous function.
Let $B$ be a separated uniform space and $D$ a directed set, let $conv(D,B)$ be the subspace of $B^{\square D}$ which contains the convergent nets. Then the map $lim : conv(D,B) \to B$ which takes a net to the point it converges to is continuous.
Let $M$ in $conv(D,B)$ and $lim M \in U \subset B$ be open. Then there is an entourage $E$ on $B$ such that $(E \circ E)[lim M] \subset U$. $E^D[M] := \{N \mid \forall i. M(i) \sim_E N(i) \} \subset conv(D,B)$ is a neighbourhood of $M$. For any $N \in E^D[M]$, we have that $lim N \in \widebar{E[lim M]}$ (as all $N(i)$ are in this closed set), so $lim N \in \widebar{E[x]} \subseteq \(E \circ E)[x] \subseteq U$. This means that $lim$ is continuous at $M$, as required.
Wikipedia, Equicontinuity
R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Kluwer Academic Publishers (2002).
Last revised on November 17, 2021 at 20:33:14. See the history of this page for a list of all contributions to it.