nLab
uniform space

Uniform spaces

Idea

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 XX consists of a collection of global binary relations ε\varepsilon which allow us to say when one point xx is “ε\varepsilon-close” to another yy. For a metric space, we have for instance relations expressed by the formula d(x,y)<εd(x, y) \lt \varepsilon, where ε\varepsilon is a positive number. For a topological group, we have relations xy 1εx y^{-1} \in \varepsilon, where ε\varepsilon is a neighborhood of the identity.

Definitions

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.

Entourage uniformities

A uniform structure, or uniformity, on a set XX consists of a collection of binary relations UX×XU \subseteq X \times X (called entourages or vicinities) satisfying some conditions. Write x Uyx \approx_U y if xx is related to yy through UU; then the conditions are the following:

  1. The equality relation Δ={(x,x):xX}\Delta = \{(x, x): x \in X\} is contained in every entourage. That is,
    U,x,x Ux. \forall U,\; \forall x,\; x \approx_U x .
  2. For every entourage UU, there exists an entourage VV such that VVUV \circ V \subseteq U, where \circ is the ordinary operation of relational composition. That is,
    U,V,x,y,z,x Vy Vzx Uz. \forall U,\; \exists V,\; \forall x, y, z,\; x \approx_V y \approx_V z \;\Rightarrow x \approx_U z .
  3. For every entourage UU, there exists an entourage VV such that V opUV^op \subseteq U, where V opV^\op is the opposite relation to VV. That is,
    U,V,x,y,y Vxx Uy. \forall U,\; \exists V,\; \forall x, y,\; y \approx_V x \;\Rightarrow\; x \approx_U y .

    In light of axiom (6), it follows that U opU^op itself is an entourage.

  4. There exists an entourage; in light of axiom (6), it follows that the universal relation X×XX \times X is an entourage.
  5. If U,VU, V are entourages, so is some subset of UVU \cap V. In light of axiom (6), it follows that UVU \cap V is an entourage.
  6. If UU is an entourage and UVX×XU \subseteq V \subseteq X \times X, then VV is an entourage.

In constructive mathematics, it is often very useful to assume that a uniform structure is located:

  • For every entourage UU, there exists an entourage VV such that ¬VU=X×X\neg{V} \cup U = X \times X, where ¬V\neg{V} is the complement of VV. That is,
    U,V,x,y,x Uyx Vy. \forall U,\; \exists V,\; \forall x, y,\; x \approx_U y \;\vee\; x \mathbin{&#x2249;}_V y .

(With excluded middle, we can take VV to be UU itself, so every uniform structure is located in classical mathematics.)

A set equipped with a uniform structure is called a uniform space.

A collection of entourages satisfying (1–5) is a base for a uniformity; a base is precisely what generates a uniformity by taking the upward closure. A collection satisfying (1–3) is a preuniformity; slightly more generally, we can replace each VV in the statement of these axioms with a finite intersection V 1V nV_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.

Covering uniformities

An equivalent way to characterize a uniform space is by its collection of uniform covers. Here a cover of a set XX is a collection CP(X)C \subseteq P(X) with union XX. For covers C iC_i, we define:

  • C 1C_1 refines C 2C_2, written C 1C 2C_1 \prec C_2, if every element of C 1C_1 is a subset of some element of C 2C_2.

  • C 1C 2{AB|AC 1,BC 2}C_1 \wedge C_2 \coloneqq \{ A \cap B \;|\; A \in C_1, B \in C_2 \}; this is also a cover.

  • For AXA \subseteq X, C[A]{BC|ABC[A] \coloneqq \bigcup \{ B \in C \;|\; A \cap B is inhabited }.

  • C *{C[A]|AC}}C^* \coloneqq \{ C[A] \;|\; A \in C\}.

We now define a covering uniformity on XX to be a collection of covers, called uniform covers, such that

  1. If CC is a uniform cover, there exists a uniform cover CC' such that (C) *C(C')^* \prec C.

  2. There exists a uniform cover; in light of axiom (4), it follows that the cover {X}\{X\} is a uniform cover.

  3. If C 1,C 2C_1, C_2 are uniform covers, so is some cover that refines C 1C 2C_1 \wedge C_2. In light of axiom (4), it follows that C 1C 2C_1 \wedge C_2 is a uniform cover.

  4. If CC is a uniform cover and CCC \prec C', then CC' 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), and then anything satisfying (1–3) is a base.

In constructive mathematics, a covering uniformity is located iff:

  • If CC is a uniform cover, there exists a uniform cover CC' such that, for any two points x,yx, y, either x,yAx, y \in A for some ACA \in C or x,yBx, y \in B for no BCB \in C'.

(With excluded middle, we can take CC' to be CC, so this is classically trivial.)

If XX 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]|xX}\{ U[x] \;|\; x \in X\} for some entourage UU, where U[x]{y|x Uy}U[x] \coloneqq \{ y \;|\; x \approx_U y \}. Note that this does not mean that a uniform cover “consists of UU-sized sets” but only that it contains a subcover consisting of sets “no smaller than UU”.

Conversely, given a covering uniformity, we define a base of entourages to consist of sets of the form {A×A|AC}\bigcup \{ A \times A \;|\; A \in C\} for CC a uniform cover. That is, for each cover CC, we have a basic entourage C\approx_C such that x Cyx \approx_C y iff x,yAx, y \in A for some ACA \in C. This defines a bijection between entourage uniformities and covering uniformities.

Further definitions

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 xx to be sets of the form

U[x]{y|x Uy} U[x] \coloneqq \{ y \;|\; x \approx_U y \}

where UU 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 XX and YY, a function f:XYf\colon X \to Y is said to be uniformly continuous if for every entourage VV of YY, (f×f) 1(V)(f \times f)^{-1}(V) is an entourage of XX. 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 XX is compact). There is an obvious concrete category UnifUnif 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 XX consists of a directed set DD and a function f:DXf\colon D \to X such that for every entourage UU, there exists NDN \in D such that f(m) Uf(n)f(m) \approx_U f(n) whenever m,nNm, n \geq N. (One can similarly define a Cauchy filter. This definition makes a uniform space into a Cauchy space.)

  • A Cauchy net f:DXf\colon D \to X converges to x:Xx\colon X if for every entourage UU, there exists NDN \in D such that nNx Uf(n)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 XX is complete if every Cauchy net/filter in XX converges (not necessarily to a unique point).

  • A uniform space XX is Hausdorff or separated if every convergent net/filter converges to a unique point, or equivalently if x=yx = y whenever x Uyx \approx_U y for every entourage UU. (This is a purely topological concept.)

Every uniform space XX admits a Hausdorff completion X¯\overline{X}, i.e., there is a uniformly continuous map XX¯X \to \overline{X} (an embedding if XX is Hausdorff, dense? in any case), characterized by the following universal property:

  • Every uniformly continuous map f:XYf\colon X \to Y to a complete Hausdorff uniform space YY extends uniquely to a uniformly continuous map f¯:X¯Y\overline{f}\colon \overline{X} \to Y.

In short, the category of complete Hausdorff uniform spaces is a reflective subcategory of UnifUnif.

One can also define a (not necessarily Hausdorff) completion of XX by replacing the image of XX in X¯\overline{X} by XX itself, but this does not have as nice properties; in particular, complete uniform spaces do not form a reflective subcategory of UnifUnif.

Every uniform space also has an underlying proximity (defined there), and the resulting functor UnifProxUnif \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.

Examples

Every metric space (or more generally any pseudometric space) is a uniform space, with a base of uniformities indexed by positive numbers ϵ\epsilon. (You can even get a countable base, for example by using only those ϵ\epsilon equal to 1/n1/n for some integer nn.) Define x ϵyx \approx_\epsilon y to mean that d(x,y)<ϵd(x,y) \lt \epsilon (or d(x,y)ϵd(x,y) \leq \epsilon if you prefer). Then axiom (2) may be proved by using ϵ/2\epsilon/2; similarly, every metric space is located in constructive mathematics, which may be proved by using any positive number less than ϵ\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 UU 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 Uyx \approx _U y to mean that xyUx \in y U (or yxUy \in x U for the other way). Then axiom (2) may be proved by invoking the continuity of multiplication; constructively, we cannot prove that every topological group is located, although this 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.

Basic Results

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 0T_0 (the topology distinguishes points), T 1T_1 (points are closed), T 2T_2 (Hausdorff), T 3T_3 (regular Hausdorff), T 312T_{3\frac1{2}} (completely regular Hausdorff). Of course, this follows from the fact that it is completely regular.

  • The category UnifUnif 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 4T_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)=0d(x, y) = 0 implies x=yx = y):

  • Every uniform space embeds uniformly in a product of pseudometric spaces. A uniform space whose uniformity admits a countable subbase is uniformly isomorphic to a pseudometric space (and hence to a metric space if the uniform space is separated, in which case the uniform space is called metrisable).

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:

  • A compact Hausdorff space XX admits a unique uniformity whose corresponding topology is the topology of XX. The entourages of this uniformity are precisely the neighborhoods of the diagonal in X×XX \times X.

It follows then that if X,YX, Y are uniform spaces with XX compact Hausdorff and if f:XYf\colon X \to Y is continuous, then ff is uniformly continuous.

Variations

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):

  • If x Uyx \approx_U y for every entourage UU, then x=yx = y.

This makes the discussion of completions slightly simpler.

If the symmetry axiom (3) 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 XX and a collection 𝒟\mathcal{D} of pseudometrics on XX; one usually requires 𝒟\mathcal{D} to be a filter. A gauge space defines a uniform space (necessarily located) by taking one basic entourage for each pseudometric in 𝒟\mathcal{D} and each positive number ϵ\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 MetMet (of metric spaces and short maps) and UnifUnif (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 don't require the uniform space to be located). 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.

Motivation for the axioms

The really critical axioms are (1–3): a collection of binary relations which satisfies those three axioms is a subbase (although not every subbase takes this form), 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 (2), which may be called an “ε2\frac{\varepsilon}{2}” principle. It generalizes a principle familiar from analysis in metric spaces, where one establishes d(x,z)<εd(x, z) \lt \varepsilon by showing there exists yy such that d(x,y)<ε2d(x, y) \lt \frac{\varepsilon}{2} and d(y,z)<ε2d(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 locatedness, which may similarly be called a “something less than ε\varepsilon” principle. (That is, for any ε\varepsilon there is an ε\varepsilon' such that any two points are either ε\varepsilon-close or ε\varepsilon'-far; classically we may take ε\varepsilon' to be ε\varepsilon, but constructively it's better to think of ε<ε\varepsilon' \lt \varepsilon.) This can actually be combined with axiom (2) into a single statement, as you might expect since ε2<ε\frac{\varepsilon}{2} \lt \varepsilon, but that makes the intuition less clear.

Axiom (1) is a nullary version of axiom (2); together they prove that, given any entourage UU and any integer n0n \geq 0, there exists an entourage VV whose nn-fold composite is contained in UU. The symmetry axiom (3) then allows one to take the opposite of VV 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: xxx \approx x and
  • symmetric: xyx \approx y iff yxy \approx x, but
  • NOT transitive: xyzx \approx y \approx z does not necessarily mean that xzx \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 ε2\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 (3) 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 uniformly continuous either way) to be equal.

Categorical interpretation

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 𝟚\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=𝟚MatRel = \mathbb{2} Mat. A quasi-uniform space can then be identified with a monoid in the bicategory ProRelPro Rel, whose hom-categories are the categories of pro-objects in the hom-categories of RelRel, 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

proarrowmonadpro-monadsymmetric versions
binary relationpreorderquasiuniformitysymmetric relationequivalence relationuniformity
binary function to [0,)[0,\infty)quasipseudometricquasiprometricsymmetric binary functionpseudometricprometric
topogenyquasiproximitysyntopogenysymmetric topogenyproximitysymmetric syntopogeny

References

  • 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

category: topology

Revised on August 15, 2014 02:56:51 by Toby Bartels (75.88.46.170)