CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Uniform spaces were invented by André Weil, to capture a general notion of space for which it makes sense to speak of uniformly continuous maps. Such spaces include (pseudo)metric spaces and topological groups.
A uniform structure or uniformity on a set $X$ consists of a collection of global binary relations $\varepsilon$ which allow us to say when one point $x$ is “$\varepsilon$-close” to another $y$. For a metric space, we have for instance relations expressed by the formula $d(x, y) \lt \varepsilon$, where $\varepsilon$ is a positive number. For a topological group, we have relations $x y^{-1} \in \varepsilon$, where $\varepsilon$ is a neighborhood of the identity.
The definition described above is based on entourages; it was the original one published in Bourbaki and is still most commonly seen today. There is an equivalent (but less directly generalisable) definition based on uniform covers. The relationship with gauge spaces (defined below) also allows for another definition.
A uniform structure, or uniformity, on a set $X$ consists of a collection of binary relations $U \subseteq X \times X$ (called entourages or vicinities) satisfying some conditions. Write $x \approx_U y$ if $x$ is related to $y$ through $U$; then the conditions are the following:
In light of axiom (6), it follows that $U^op$ itself is an entourage.
In constructive mathematics, we also need this condition:
We will call this axiom (0) when convenient. (With excluded middle, we can take $V$ to be $U$ itself, so this is classically trivial.)
A set equipped with a uniform structure is called a uniform space.
A collection of entourages satisfying (0–5) is a base for a uniformity; a base is precisely what generates a uniformity by taking the upward closure. A collection satisfying (0–3) is a preuniformity; slightly more generally, we can replace each $V$ in the statement of these axioms with a finite intersection $V_1 \cap \cdots \cap V_n$ of entourages to get the concept of a subbase for a uniformity. A subbase is precisely what generates a base by closing up under finite intersections and precisely what generates a uniformity by closing up under finite intersections and taking the upward closure.
An equivalent way to characterize a uniform space is by its collection of uniform covers. Here a cover of a set $X$ is a collection $C \subseteq P(X)$ with union $X$. For covers $C_i$, we define:
$C_1$ refines $C_2$, written $C_1 \prec C_2$, if every element of $C_1$ is a subset of some element of $C_2$.
$C_1 \wedge C_2 \coloneqq \{ A \cap B \;|\; A \in C_1, B \in C_2 \}$; this is also a cover.
For $A \subseteq X$, $C[A] \coloneqq \bigcup \{ B \in C \;|\; A \cap B$ is inhabited }
.
$}C^* \coloneqq \{ C[A] \;|\; A \in C\}$.
We now define a covering uniformity on $X$ to be a collection of covers, called uniform covers, such that
If $C$ is a uniform cover, there exists a uniform cover $C'$ such that $(C')^* \prec C$.
There exists a uniform cover; in light of axiom (4), it follows that the cover $\{X\}$ is a uniform cover.
If $C_1, C_2$ are uniform covers, so is some cover that refines $C_1 \wedge C_2$. In light of axiom (4), it follows that $C_1 \wedge C_2$ is a uniform cover.
If $C$ is a uniform cover and $C \prec C'$, then $C'$ is a uniform cover.
The axioms (2–4) here roughly correspond (respectively) to the axioms (4–6) in the entourage definition. Axiom (1) takes on all of the real work; any collection of covers that satisfies it may be called a subbase (but not corresponding directly to a subbase in the previous definition).
In constructive mathematics, we also want to add an axiom (0):
(With excluded middle, we can take $C'$ to be $C$, so this is classically trivial.)
Is there a more natural way to state this in this context?
If $X$ is a uniform space defined in terms of entourages, we give it a covering uniformity by declaring a cover to be uniform if it is refined by $\{ U[x] \;|\; x \in X\}$ for some entourage $U$, where $U[x] \coloneqq \{ y \;|\; x \approx_U y \}$. Note that this does not mean that a uniform cover “consists of $U$-sized sets” but only that it contains a subcover consisting of sets “no smaller than $U$”.
Conversely, given a covering uniformity, we define a base of entourages to consist of sets of the form $\bigcup \{ A \times A \;|\; A \in C\}$ for $C$ a uniform cover. That is, for each cover $C$, we have a basic entourage $\approx_C$ such that $x \approx_C y$ iff $x, y \in A$ for some $A \in C$. This defines a bijection between entourage uniformities and covering uniformities.
We give these in terms of entourages, but they could also be given directly in terms of uniform covers if desired.
The uniform topology induced by a uniformity is defined by taking the neighborhoods of a point $x$ to be sets of the form
where $U$ is an entourage. (Recall that a subset is open iff it is a neighborhood of every point that it contains.) We point out that different uniformities may give rise to the same topology (just as different metrics, even uniformly inequivalent ones, may give rise to the same topology).
Given uniform spaces $X$ and $Y$, a function $f\colon X \to Y$ is said to be uniformly continuous if for every entourage $V$ of $Y$, $(f \times f)^{-1}(V)$ is an entourage of $X$. Clearly, uniformly continuous functions are continuous with respect to the corresponding uniform topologies, but the converse is false (although see below the discussion in the case where $X$ is compact). There is an obvious concrete category $Unif$ of uniform spaces and uniformly continuous maps.
One feature of uniform space theory which is not available for general topological spaces is the possibility of taking (Cauchy) completions. The relevant definitions are straightforward:
A Cauchy net in a uniform space $X$ consists of a directed set $D$ and a function $f\colon D \to X$ such that for every entourage $U$, there exists $N \in D$ such that $f(m) \approx_U f(n)$ whenever $m, n \geq N$. (One can similarly define a Cauchy filter. This definition makes a uniform space into a Cauchy space.)
A Cauchy net $f\colon D \to X$ converges to $x\colon X$ if for every entourage $U$, there exists $N \in D$ such that $n \geq N \;\Rightarrow\; x \approx_U f(n)$. (This makes sense for arbitrary nets or filters, but it can be proved that any convergent net is Cauchy. This definition makes a uniform space into a convergence space.)
A uniform space $X$ is complete if every Cauchy net/filter in $X$ converges (not necessarily to a unique point).
A uniform space $X$ is Hausdorff or separated if every convergent net/filter converges to a unique point. (This is a purely topological concept.)
Every uniform space $X$ admits a Hausdorff completion $\overline{X}$, i.e., there is a uniformly continuous map $X \to \overline{X}$ (an embedding if $X$ is Hausdorff, dense? in any case), characterized by the following universal property:
In short, the category of complete Hausdorff uniform spaces is a reflective subcategory of $Unif$.
One can also define a (not necessarily Hausdorff) completion of $X$ by replacing the image of $X$ in $\overline{X}$ by $X$ itself, but this does not have as nice properties; in particular, complete uniform spaces do not form a reflective subcategory of $Unif$.
Every uniform space also has an underlying proximity (defined there), and the resulting functor $Unif \to Prox$ has a fully faithful right adjoint which identifies proximity spaces with uniform spaces that are totally bounded.
Uniform spaces can also be identified with syntopogenous spaces that are both perfect and symmetric; see syntopogenous space.
Every metric space (or more generally any pseudometric space) is a uniform space, with a base of uniformities indexed by positive numbers $\epsilon$. (You can even get a countable base, for example by using only those $\epsilon$ equal to $1/n$ for some integer $n$.) Define $x \approx_\epsilon y$ to mean that $d(x,y) \lt \epsilon$ (or $d(x,y) \leq \epsilon$ if you prefer). Then axiom (3) may be proved by using $\epsilon/2$; axiom (0) may be proved constructively by using any positive number less than $\epsilon$ and applying the comparison law. (The other axioms are easy.) Every quasi(pseudo)metric space is a quasiuniform space in the same way. We can also generalise from metric spaces to gauge spaces; see under Variations below.
Every topological group is a uniform space, with a base of uniformities indexed by neighbourhoods $U$ of the identity element, in two ways: left and right. (These two ways agree for abelian groups, of course; they also agree for compact groups, by the general theorem below for uniformities on compact spaces. I wonder if that has anything to do with Haar measure?) In particular, any Banach space or Lie group is a uniform space. Define $x \approx _U y$ to mean that $x \in y U$ (or $y \in x U$ for the other way). Then axiom (3) may be proved by invoking the continuity of multiplication; axiom (0) cannot be proved constructively in general, although it can be proved for the classical examples of Lie groups and topological vector spaces (TVSs).
These are in a way the motivating examples. The theory of uniformly continuous maps was developed first for metric spaces, but it was noticed that, for a metrisable TVS, it could be described entirely in terms of the topology and the addition, which immediately generalised to non-metrisable TVSs. The theory of uniform spaces covers both of these (and their generalisations to pseudometric spaces and topological groups) at once.
A first wave of results concerns separation axioms:
As a matter of fact, a significant theorem is that a topology (on a given set) is the uniform topology for some uniformity if and only if it is completely regular. See also the discussion below on the relation with metric and pseudometric spaces.
For a uniform space, the following separation conditions are equivalent: $T_0$ (the topology distinguishes points), $T_1$ (points are closed), $T_2$ (Hausdorff), $T_3$ (regular Hausdorff), $T_{3\frac1{2}}$ (completely regular Hausdorff). Of course, this follows from the fact that it is completely regular.
The category $Unif$ of uniform spaces admits arbitrary small products (which are preserved by the forgetful functors to Top and to Set). Hence it is not generally true that uniform spaces are normal (so that separated ones would be $T_4$), because for instance an uncountable power of the real line (with its usual topology) is not a normal space.
A second wave of results relates uniform spaces to pseudometric spaces (like metric spaces, but dropping the separation axiom that $d(x, y) = 0$ implies $x = y$):
By this result, the results mentioned above on completions of uniform spaces may be proved by appeal to similar results for (pseudo)metric spaces.
Finally, we mention the special case of compact spaces:
It follows then that if $X, Y$ are uniform spaces with $X$ compact Hausdorff and if $f\colon X \to Y$ is continuous, then $f$ is uniformly continuous.
Some authors insist that a uniform space must be separated; this can be arranged directly in the definition (that is without reference to the uniform topology or to the concept of convergent net/filter) by adding the following axiom, a sort of converse axiom (1):
This makes the discussion of completions slightly simpler.
If the symmetry axiom (2) is dropped, then the result is a quasiuniform space. Quasiuniform spaces are related to quasi(pseudo)metrics in the same way as uniform spaces are related to (psuedo)metrics. Perhaps surprisingly, every topological space is quasiuniformisable. (There does not seem to be a way to define quasiuniform spaces in terms of (quasi)uniform covers.)
A gauge space consists of a set $X$ and a collection $\mathcal{D}$ of pseudometrics on $X$; one usually requires $\mathcal{D}$ to be a filter. A gauge space defines a uniform space by taking one basic entourage for each pseudometric in $\mathcal{D}$ and each positive number $\epsilon$; conversely, every uniform space arises in this way, with the pseudometrics in the gauge being those that are uniformly continuous as maps on the product space. However, gauge spaces form a category with a stricter notion of morphism, in which the categories $Met$ (of metric spaces and short maps) and $Unif$ (of uniform spaces and uniformly continuous maps) are both full subcategories. A quasigauge space consists of a set and a collection of quasipseudometrics; every quasiuniform space arises from a quasigauge space.
In weak foundations of mathematics, the theorems above may not be provable. In particular, the theorem that every uniform space arises from a gauge space is equivalent (internal to an arbitrary topos with a natural numbers object) to dependent choice (plus excluded middle if you leave out axiom 0). If the concept is to be applied to analysis, then it may be best to define a uniform space as a gauge space satisfying a saturation condition.
There is also a “pointless” notion of uniform space, called a uniform locale.
The really critical axioms are (1–3): a collection of binary relations which satisfies those three axioms is a subbase, from we can generate a uniform structure straightforwardly. Indeed, (4–6) simply state that the entourages form a filter, so generating a uniformity from a base or subbase is simply the usual generation of a filter from a base or subbase.
We draw particular attention to axiom (3), which may be called an “$\frac{\varepsilon}{2}$” principle. It generalizes a principle familiar from analysis in metric spaces, where one establishes $d(x, z) \lt \varepsilon$ by showing there exists $y$ such that $d(x, y) \lt \frac{\varepsilon}{2}$ and $d(y, z) \lt \frac{\varepsilon}{2}$, and applying the triangle inequality. The utility of this principle for metric spaces, extrapolated in this way, gives uniform spaces much of their power.
For full power in constructive mathematics, we also need axiom (0), which may similarly be called a “something less than $\varepsilon$” principle. (That is, for any $\varepsilon$ there is an $\varepsilon'$ such that any two points are either $\varepsilon$-close or $\varepsilon'$-far; classically we may take $\varepsilon'$ to be $\varepsilon$, but constructively it's better to think of $\varepsilon' \lt \varepsilon$.) Axioms (0) and (3) can actually be combined into a single statement, as you might expect since $\frac{\varepsilon}{2} \lt \varepsilon$, but that makes the intuition less clear.
Axiom (1) is a nullary version of axiom (3); together they prove that, given any entourage $U$ and any integer $n \geq 0$, there exists an entourage $V$ whose $n$-fold composite is contained in $U$. The symmetry axiom then allows one to take the opposite of $V$ at any point in the composite as well.
Altogether, these may be seen as axiomatising the notion of approximate equivalence. If $\approx$ is an approximate equivalence relation, then we might expect it to be * reflexive: $x \approx x$ and * symmetric: $x \approx y$ iff $y \approx x$, but * NOT transitive: $x \approx y \approx z$ does not necessarily mean that $x \approx z$.
One could stop there, but this is not a very useful notion of approximation. Instead we generalise to a family of approximate equivalence relations and impose the $\frac{\varepsilon}{2}$ principle to allow them to be used. This is nearly the definition of uniform space; in particular, the axiom (1) states precisely that each entourage is reflexive. The symmetry axiom in the standard definition is weaker than requiring each individual entourage to be symmetric, but that is not an essential change; every uniformity has a base consisting of its symmetric entourages. The final three axioms have already been explained as a closure condition; they force equivalent uniformities on a given set (in the sense that the identity function on the set is uniformaly continuous either way) to be equal.
The idea that a uniformity is an “approximate equivalence relation” can be made precise as follows. A preorder is the same as a category enriched over the poset $\mathbb{2}$ of truth values, and it is an equivalence relation if and only if this category is symmetric. In fancier language, a preorder is a monoid (or monad) in the bicategory $Rel = \mathbb{2} Mat$. A quasi-uniform space can then be identified with a monoid in the bicategory $Pro Rel$, whose hom-categories are the categories of pro-objects in the hom-categories of $Rel$, aka filters. Of course, it is a uniform space just when it is also symmetric. See also prometric space.
In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monoids in a double category or equipment rather than merely a bicategory.
Generalized uniform structures
A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Actualités Sci. Ind. 551, Paris, (1937).
John Kelley, General Topology, GTM 27, 1955.
Eric Schechter, Handbook of Analysis and its Foundations.
Warren Page, Topological Uniform Structures, Dover
I.M. James, Topologies and Uniformities, Springer
Norman Howes, Modern analysis and topology, Springer
P. Samuel, Ultrafilters and compactifications of uniform spaces, Trans. Amer. Math. Soc. 64 (1948) 100-132
J. R. Isbell, Uniform spaces, Math. Surveys 12, Amer. Math. Soc. 1964