nLab
net

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Analysis

Contents

Idea

A net in a set XX is a function from a directed set DD to XX. Special cases of nets are sequences, for which D= D = \mathbb{N}_{\leq} is the natural numbers. Regarded as a generalization of sequences, nets are used in topology for formalization of the concept of convergence.

Nets are also called Moore–Smith sequences and are equivalent (in a certain sense) to proper filters (def. 8 below), their eventuality filters (def. 9 below).

The concept of nets is motivated from the fact that where plain sequences detect topological properties in metric spaces, in generally they fail to do so in more general topological spaces. For example sequentially compact metric spaces are equivalently compact metric spaces, but for general topological spaces being sequentially compact neither implies nor is implied by being compact (see at sequentially compact space Examples and counter-examples).

Inspection of these counter-examples reveals that the problem is that sequences indexed by the natural numbers may be “too short” in that they cannot go deep enough into uncountable territory, and they are “too slim” in that they proceed to their potential limiting point only from one direction, instead of from many at once. The use of general directed sets for nets in place of just the natural numbers for sequences fixes these two issues.

And indeed, as opposed to sequences, nets do detect

  1. the topology on general topological spaces (prop. 2 below),

  2. the continuity of functions between them (prop. 3 below),

  3. the Hausdorff property (prop. 4 below),

  4. compactness (prop. 5 below).

While the concept of nets is similar to that of sequences, one gets a cleaner theory still by considering not the nets themselves but their “filters of subsets which they eventually meet” (def. 3 below), called their eventuality filters (def. 9 below). For example equivalent filters are equal (in contrast to nets) and (unless in predicative mathematics) the set of filters on a set XX is small (not a proper class).

Definitions

Directed sets

Definition

(directed set)

A directed set is

such that

  • every finite subset has an upper bound, hence for any a,bDa,b \in D there exists cDc \in D with aca \leq c and bcb \leq c.
Example

(directed set of natural numbers)

The natural numbers \mathbb{N} with their canonical lower-or-equal relation \leq form a directed set (def. 1).

The key class of examples of nets, underlying their relation to topology (below) is the following:

Example

(directed set of neighbourhoods)

Let (X,τ)(X, \tau) be a topological space and let xXx \in X be an element of the underlying set. Then then set of (Nbhd X(x) )(Nbhd_X(x)_{\supset}) neighbourhoods of xx, ordered by reverse inclusion, is a directed set (def. 1).

Example

Let A A_{\geq} and B B_{\geq} be two directed sets (def. 1). Then the Cartesian product A×BA \times B of the underlying sets becomes itself a directed set by setting

((a 1,b 1)(a 2,b 2))((a 1a 2)and(b 1b 2)). \left( (a_1, b_1) \leq (a_2, b_2) \right) \,\coloneqq\, \left( \left( a_1 \leq a_2\right) \,\text{and}\, \left( b_1 \leq b_2 \right) \right) \,.

Nets

Definition

For XX a set, then a net in XX is

  1. a directed set AA (def. 1), called the index set,

  2. a function ν:AX\nu \colon A \to X from (the underlying set of) AA to XX.

We say that AA indexes the net.

Example

(sequences are nets)

A sequence is a net (def. 2) whose directed set of indices is the natural numbers (,)(\mathbb{N}, \leq) (example 1).

Remark

Although the index set AA in def. 2, being a directed set, is equipped with a preorder, the function ν:AX\nu \colon A \to X is not required to preserve this in any way. This forms an exception to the rule of thumb that a preordered set may be replaced by its quotient poset.

You can get around this if you instead define a net in XX as a multi-valued function from a partially ordered directed set AA to XX. Although there is not much point to doing this in general, it can make a difference if you put restrictions on the possibilities for AA, in particular if you consider the definition of sequence. In some type-theoretic foundations of mathematics, you can get the same effect by defining a net to be an ‘operation’ (a prefunction, like a function but not required to preserve equality).

Definition

(eventually and frequently)

Consider net ν:AX\nu \colon A \to X (def. 2), and given a subset SXS \subset X. We say that

  1. ν\nu is eventually in SS if there exists iAi \in A such that ν jS\nu_j \in S for every jij \ge i.

  2. ν\nu is frequently in SS if for every index iAi \in A, then ν jS\nu_j \in S for some jij \ge i.

Remark

Sometimes one says ‘infinitely often’ in place of ‘frequently’ in def. 3 and even ‘cofinitely often’ in place of ‘eventually’; these derive from the special case of sequences, where they may be taken literally.

Definition

(convergence of nets)

Let (X,τ)(X,\tau) be a topological space, and let ν:AX\nu \colon A \to X be a net in the underlying set (def. 2).

We say that the net ν\nu

  1. converges to an element xXx \in X if given any neighbourhood UU of xx, ν\nu is eventually in UU (def. 3); such xx is called a limit point of the net;

  2. clusters at xx if, for every neighbourhood UU of xx, ν\nu is frequently in UU (also def. 3); such xx is called a cluster point of the net.

Remark

Beware that limit points of nets, according to def. 4, need not be unique. They are guaranteed to be unique in Hausdorff spaces, see prop. 4 below.

Subnets

The definition of the concept of sub-nets of a net requires some care. The point of the definition is to ensure that prop. 5 below becomes true, which states that compact spaces are equivalently those for which every net has a converging subnet.

There are several different definitions of ‘subnet’ in the literature, all of which intend to generalise the concept of subsequences. We state them now in order of increasing generality. Note that it is Definition 7 which is correct in that it corresponds precisely to refinement of filters. However, the other two definitions (def. 5, def. 6) are sufficient (in a sense made precise by theorem 1 below) and may be easier to work with.

Definition

(Willard, 1970).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

We have a function f:BAf\colon B \to A such that

  • ff maps xx to yy (that is, for every βB\beta \in B, y β=x f(β)y_{\beta} = x_{f(\beta)});
  • ff is monotone (that is, for every β 1β 2B\beta_1 \geq \beta_2 \in B, f(β 1)f(β 2)f(\beta_1) \geq f(\beta_2));
  • ff is cofinal (that is, for every αA\alpha \in A there is a βB\beta \in B such that f(β)αf(\beta) \geq \alpha).
Definition

(Kelley, 1955).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

We have a function f:BAf\colon B \to A such that

  • ff maps xx to yy (that is, for every βB\beta \in B, y β=x f(β)y_{\beta} = x_{f(\beta)});
  • ff is strongly cofinal (that is, for every αA\alpha \in A there is a βB\beta \in B such that, for every β 1βB\beta_1 \geq \beta \in B, f(β 1)αf(\beta_1) \geq \alpha).
Remark

Notice that the function ff in definitions 5 and 6 is not required to be an injection, and it need not be. As a result, a sequence regarded as a net in general has more sub-nets than it has sub-sequences.

Definition

(Smiley, 1957; Årnes & Andenæs, 1972).

Given a net (x α)(x_{\alpha}) with index set AA, and a net (y β)(y_{\beta}) with an index set BB, we say that yy is a subnet of xx if:

The eventuality filter of yy (def. 9) refines the eventuality filter of xx. (Explicitly, for every αA\alpha \in A there is a βB\beta \in B such that, for every β 1βB\beta_1 \geq \beta \in B there is an α 1αA\alpha_1 \geq \alpha \in A such that y β 1=x α 1y_{\beta_1} = x_{\alpha_1}.)

The equivalence between these definitions is as follows:

Theorem

(Schechter, 1996).

  1. If yy is a (5)-subnet of xx, then yy is also a (6)-subnet of xx, using the same function ff.
  2. If yy is a (6)-subnet of xx, then yy is also a (7)-subnet of xx.
  3. If yy is a (7)-subnet of xx, then there is some net zz such that
    • zz is equivalent to yy in the sense that yy and zz are (7)-subnets of each other, and
    • zz is a (5)-subnet of xx, using some function.

So from the perspective of definition (7), there are enough (5)-subnets and (6)-subnets, up to equivalence.

Eventuality filters

Recall that:

Definition

(filter)

Given a set XX then a set of subsets of XX, hence a subset of the power set

P(X) \mathcal{F} \subset P(X)

is called a filter of subsets if it is closed under intersections and under taking supersets.

The filter \mathcal{F} is called proper if each set in it is inhabited.

Definition

(eventuality filter)

Let XX be a set and let ν:DX\nu \colon D \to X be a net in XX (def. 2).

The eventuality filter ν\mathcal{F}_\nu of the net ν\nu is the filter (def. 8) onsisting of the subsets that ν\nu is eventually in, according to def. 3.

((UX) ν)(νis eventually inU). \left( (U \subset X) \in \mathcal{F}_\nu \right) \,\Leftrightarrow\, \left( \nu \, \text{is eventually in}\, U \right) \,.
Remark

(equivalence of nets)

Two nets are to be considered equivalent if they have the same eventuality filter according to def. 9. By def. 7 and theorem 1, this means equivalently that they are both subnets of each other.

In particular, equivalent nets define the same logical quantifiers (see below) and are therefore indeed equivalent for the application to topology (see below).

(Of course, it is possible to distinguish them by using the standard logical quantifiers instead.)

Conversely, every filter is the eventuality filter of some net:

Definition

(nets from filters)

Let XX be a set and let P(X)\mathcal{F} \subset P(X) be a filter of subsets of XX (def. 8). Ss Consider the disjoint union U\underset{U \in \mathcal{F}}{\sqcup} of subsets in \mathcal{F}, hence the set whose elements are pairs of the form (U,x)(U,x), where xUx \in U \in \mathcal{F}. Equipped with the ordering

((U,x)(V,y))(UV)AAAregardless ofxandy \left( (U,x) \geq (V,y) \right) \,\Leftrightarrow\, \left( U \subset V \right) \phantom{AAA} \text{regardless of}\, x\, \text{and} \, y

the fact that \mathcal{F} is a proper filter implies that this is a directed set according to def. 1. (It is actually enough to use only a base of the filters).

Then the filter net ν F\nu_F of \mathcal{F} is the net on XX (def. 2) given by

(UU) ν X (U,x) AAA x. \array{ \left( \underset{U \in \mathcal{F}}{\sqcup} U \right)_{\supset} &\overset{\nu_{\mathcal{F}}}{\longrightarrow}& X \\ (U,x) &\overset{\phantom{AAA}}{\mapsto}& x } \,.
Proposition

Given a set XX and a filter of subsets P(X)\mathcal{F} \subset P(X) (def. 8), then \mathcal{F} is the eventuality filter (def. 9) of its filter net (def. 10).

Properties

Relation to topology

We discuss that nets detect:

  1. the topology on general topological spaces (prop. 2 below),

  2. the continuity of functions between them (prop. 3 below),

  3. the Hausdorff property (prop. 4 below),

  4. compactness (prop. 5 below).

Proposition

(topology detected by nets)

Using the axiom of choice then:

Let (X,τ)(X, \tau) be a topological space. Then a subset (UX)(U \subset X) is open in XX (is an element of τP(X)\tau \subset P(X)) precisely if its complement X\SX \backslash S is a closed subset as seen not just by sequences but by nets, in that no net with elements in X\SX\backslash S, ν:AX\SX\nu \colon A \to X\backslash S \hookrightarrow X, converges to an element in SS.

Proof

In one direction, let SXS \subset X be open, and consider a net ν:AX\SX\nu \colon A \to X \backslash S \subset X. We need to show that for every point xSx \in S, xx is not a limiting point of the net.

But by assumption then SS is a neighbourhood of xx which does not contain any element of the net, and so by definition of convergence it is not a limit of this net.

Conversely, let SXS \subset X be a subset that is not open. We need to show that then there exists a net ν:AX\SX\nu \colon A \to X\backslash S \subset X that converges to a point in SS.

For xXx \in X, consider the directed set Nbhd X(x) Nbhd_X(x)_{\supset} of open neighbourhoods of this element (example 2). Now the fact that the set SS is not open means that there exists an element sSXs \in S \subset X such that every open neighbourhood UU of ss intersects X\SX \backslash S. This means that we may choose elements x UU(X\S)x_U \in U \cap (X \backslash S), and hence define a net

Nbhds X(s) ν X\SX U x U. \array{ Nbhds_X(s) &\overset{\nu}{\longrightarrow}& X \backslash S \subset X \\ U &\mapsto& x_U } \,.

But by construction this net has the property that for every neighbourhood VV of ss there exists UNbhd X(s)U \in Nbhd_X(s) such that for all UUU' \subset U then x UVx_{U'} \in V, namely U=VU = V. Hence the net converges to ss.

Proposition

(continuous functions detected by nets)

Assuming excluded middle, then

Let (X,τ X)(X,\tau_X) and (Y,τ Y)(Y,\tau_Y) be two topological space. Then a function f:XYf \colon X \to Y between their underlying sets is continuous precisely if for every net ν:AX\nu \colon A \to X that converges to some limit point xXx \in X (def. 4), the image net fνf\circ \nu converges to f(y)Yf(y)\in Y.

Proof

In one direction, suppose that f:XYf \colon X \to Y is continuous, and that ν:AX\nu \colon A \to X converges to some xXx \in X. We need to show that fcitcνf \citc \nu converges to f(y)Yf(y) \in Y, hence that for every neighbourhood U f(y)YU_{f(y)} \subset Y there exists iAi \in A such that f(ν(j))U f(y)f(\nu(j)) \in U_{f(y)} for all jij \geq i.

But since ff is continuous, the pre-image f 1(U f(y))Xf^{-1}(U_{f(y)}) \subset X is an open neighbourhood of xx, and so by the assumption that ν\nu converges there is an iAi \in A such that ν(j)f 1(U f(y))\nu(j) \in f^{-1}(U_{f(y)}) for all jij \geq i. By applying ff, this is the required statement.

Conversely, suppose that ff is not continuous, and that the net ν\nu converges to some xXx \in X. We need to show that then fνf \circ \nu does not converge to f(x)f(x). (This is the contrapositive of the reverse implication, and by excluded middle equivalent to it.)

Now that ff is not continuous means that there exists an open subset UYU \subset Y such that the pre-image f 1(U)f^{-1}(U) is not open. By prop. 2 this means that there exists a net ν\nu in X\f 1(U)X \backslash f^{-1}(U) that converges to an element xf 1(U)x \in f^{-1}(U). But this means that fνf \circ \nu is a net in the Y\UY \backslash U, which is a closed subset by the assumption that UU is open. Again by prop. 2 this means that fνf\circ \nu converges to an element in Y\UY \backslash U, and hence not to f(x)Uf(x) \in U.

Remark

It is possible to define elementary conditions on this convergence relation that characterise whether it is topological (that is whether it comes from a topology on XX), although these are a bit complicated.

By keeping only the simple conditions, one gets the definition of a convergence space; this is a more general concept than a topological space and includes many non-topological situations where we want to say that a sequence converges to some value (such as convergence in measure).

Proposition

(Hausdorff property detected by nets)

Assuming excluded middle and the axiom of choice, then:

A topological space (X,τ)(X,\tau) is Hausdorff topological space precisely if no net in XX (def. 2) converges to two distinct limit points (def. 4).

Proof

In one direction, assume that (X,τ)(X,\tau) is a Hausdorff space, and that ν:AX\nu \colon A \to X is a net in XX which has limits points x 1,x 2Xx_1, x_2 \in X. We need to show that then x 1=x 2x_1 = x_2.

Assume on the contrary that the two points were different, x 1x 2x_1 \neq x_2. By assumption of Hausdorffness, these would then have disjoint open neighbourhoods U x 1,U x 2U_{x_1}, U_{x_2}, i.e. U 1U 2=U_1 \cap U_2 = \emptyset. By definition of convergence, there would thus be a 1,a 2Aa_1, a_2 \in A such that ν a 1U x 1\nu_{a_1 \leq \bullet} \in U_{x_1} and ν a 2U x 2\nu_{a_2 \leq \bullet} \in U_{x_2}. Moreover, by the definition of directed set, this would imply a 3Aa_3 \in A with a 1,a 2a 3a_1, a_2 \leq a_3, and hence that x a 3U x 1U x 2x_{a_3 \leq \bullet} \in U_{x_1} \cap U_{x_2}. This is in contradiction to the emptiness of the intersection, and hence we have a proof by contradiction.

Conversely, assume that (X,τ)(X,\tau) is not a Hausdorff space. We need to show that then there exists a net ν\nu in XX with two distinct limit points.

That (X,τ)(X,\tau) is not Hausdorff means that there are two distinct points x 1,x 2Xx_1, x_2 \in X such that every open neighbourhood of x 1x_1 intersects every open neighbourhood of x 2x_2. Hence we may choose elements in these intersections

x U x 1,U x 2U x 1,U x 2. x_{U_{x_1}, U_{x_2}} \in U_{x_1}, U_{x_2} \,.

Consider the directed neighbourhood sets Nbhd X(x 1) Nbhd_X(x_1)_{\supset} and Nbhd X(x 2) Nbhd_X(x_2)_{\supset} of these two points (example 2) and their directed Cartesian product set (example 3) Nbhd X(x 1) ×Nbhd X(x 2) Nbhd_X(x_1)_{\supset} \times Nbhd_X(x_2)_{\supset}. The above elements then define a net

Nbhd X(x 1)×Nbhd X(x 2) ν X (U x 1,U x 2) AAA x U 1,U 2. \array{ Nbhd_X(x_1) \times Nbhd_X(x_2) &\overset{\nu}{\longrightarrow}& X \\ (U_{x_1}, U_{x_2}) &\overset{\phantom{AAA}}{\mapsto}& x_{U_1, U_2} } \,.

We conclude by claiming that x 1x_1 and x 2x_2 are both limit points of this net. We show this for x 1x_1, the argument for x 2x_2 is directly analogous:

Let U x 1U_{x_1} be an open neighbourhood of x 1x_1. We need to find an element (V 1,V 2)Nbhd X(x 1)×Nbhd X(x 2)(V_1, V_2) \in Nbhd_X(x_1) \times Nbhd_X(x_2) such that for all (W 1,W 2)(V 1,V 2)(W_1, W_2) \subset (V_1, V_2) then ν (W 1,W 2)U x 1\nu_{(W_1, W_2)} \in U_{x_1}.

Take V 1U x 1V_1 \coloneqq U_{x_1} and take V 2=XV_2 = X. Then by construction

ν (W 1,W 2) W 1W 2 V 1V 2 =U x 1X =U x 1. \begin{aligned} \nu_{(W_1, W_2)} & \in W_1 \cap W_2 \\ & \subset V_1 \cap V_2 \\ & = U_{x_1} \cap X \\ & = U_{x_1} \end{aligned} \,.
Proposition

(compact spaces are equivalently those for which every net has a converging subnet)

Assuming excluded middle and the axiom of choice, then:

A topological space (X,τ)(X,\tau) is compact precisely if every net in XX (def. 2) has a sub-net (def. 5) that converges (def. 4).

We break up the proof into that of lemmas 5 and 6:

Lemma

(in a compact space, every net has a convergent subnet)

Let (X,τ)(X,\tau) be a compact topological space. Then every net in XX has a convergent subnet.

Proof

Let ν:AX\nu \colon A \to X be a net. We need to show that there is a subnet which converges.

For aAa \in A consider the topological closures Cl(S a)Cl(S_a) of the sets S aS_a of elements of the net beyond some fixed index:

S a{ν bX|ba}X. S_a \;\coloneqq\; \left\{ \nu_b \in X \;\vert\; b \geq a \right\} \subset X \,.

Observe that the set {S aX} aA\{S_a \subset X\}_{a \in A} and hence also the set {Cl(S a)X} aA\{Cl(S_a) \subset X\}_{a \in A} has the finite intersection property, by the fact that AA is a directed set. Therefore this prop. implies from the assumption of XX being compact that the intersection of all the Cl(S a)Cl(S_a) is non-empty, hence that there is an element

xaACl(S a). x \in \underset{a \in A}{\cap} Cl(S_a) \,.

In particular every neighbourhood U xU_x of xx intersects each of the Cl(S a)Cl(S_a), and hence also each of the S aS_a. By definition of the S aS_a, this means that for every aAa \in A there exists bab \geq a such that ν bU x\nu_b \in U_x, hence that xx is a cluster point (def 4) of the net.

We will now produce a sub-net

B f A ν X \array{ B && \overset{f}{\longrightarrow} && A \\ & \searrow && \swarrow_{\nu} \\ && X }

that converges to this cluster point. To this end, we first need to build the domain directed set BB. Take it to be the sub-directed set of the Cartesian product directed set (example 3) of AA with the directed neighbourhood set Nbhd X(x)Nbhd_X(x) of xx (example 2)

BA ×Nbhd X(x) B \subset A_{\leq} \times Nbhd_X(x)_{\supset}

on those pairs such that the element of the net indexed by the first component is contained in the second component:

B{(a,U x)|ν aU X}. B \;\coloneqq\; \left\{ (a,U_x) \,\vert \, \nu_a \in U_X \right\} \,.

It is clear BB is a preordered set. We need to check that it is indeed directed, in that every pair of elements (a 1,U 1)(a_1, U_1), (a 2,U 2)(a_2, U_2) has a common upper bound (a bd,U bd)(a_{bd}, U_{bd}). Now since AA itself is directed, there is an upper bound a 3a 1,a 2a_3 \geq a_1, a_2, and since xx is a cluster point of the net there is moreover an a bda 3a 1,a 3a_{bd} \geq a_3 \geq a_1, a_3 such that ν a bdU 1U 2\nu_{a_{bd}} \in U_1 \cap U_2. Hence with U bdU 1U 2U_{bd} \coloneqq U_1 \cap U_2 we have obtained the required pair.

Next take the function ff to be given by

B f A (a,U) AAA a. \array{ B &\overset{f}{\longrightarrow}& A \\ (a, U) &\overset{\phantom{AAA}}{\mapsto}& a } \,.

This is clearly order preserving, and it is cofinal since it is even a surjection. Hence we have defined a subnet νf\nu \circ f.

It now remains to see that νf\nu \circ f converges to xx, hence that for every open neighbourhood U xU_x of xx we may find (a,U)(a,U) such that for all (b,V)(b,V) with aba \leq b and UVU \supset V then ν(f(b,V))=ν(b)U x\nu(f(b,V)) = \nu(b) \in U_x. Now by the nature of xx there exists some aa with ν aU x\nu_a \in U_x, and hence if we take UU xU \coloneqq U_x then nature of BB implies that with (b,V)(a,U x)(b, V) \geq (a,U_x) then bVU xb \in V \subset U_x.

Lemma

Assuming excluded middle, then:

Let (X,τ)(X,\tau) be a topological space. If every net in XX has a subnet that converges, then (X,τ)(X,\tau) is a compact topological space.

Proof

By excluded middle we may equivalently prove the contrapositive: If (X,τ)(X,\tau) is not compact, then not every net in XX has a convergent subnet.

Hence assume that (X,τ)(X,\tau) is not compact. We need to produce a net without a convergent subnet.

Again by excluded middle, then by this prop. (X,τ)(X,\tau) not being compact means equivalently that there exists a set {C iX} iI\{C_i \subset X\}_{i \in I} of closed subsets satisfying the finite intersection property, but such that their intersection is empty: iIC i=\underset{i \in I}{\cap} C_i = \emptyset.

Consider then P fin(I)P_{fin}(I), the set of finite subsets of II. By the assumption that {C iX} iI\{C_i \subset X\}_{i \in I} satisfies the finite intersection property, we may choose for each JP fin(I)J \in P_{fin}(I) an element

x JiJIC i. x_J \in \underset{i \in J \subset I}{\cap} C_i \,.

Now P fin(X)P_{fin}(X) regarded as a preordered set under inclusion of subsets is clearly a directed set, with an upper bound of two finite subsets given by their union. Therefore we have defined a net

P fin(X) ν X J AAA x J. \array{ P_{fin}(X)_{\subset} &\overset{\nu}{\longrightarrow}& X \\ J &\overset{\phantom{AAA}}{\mapsto}& x_J } \,.

We will show that this net has no converging subnet.

Assume on the contrary that there were a subnet

B f P fin(X) ν X \array{ B && \overset{f}{\longrightarrow} && P_{fin}(X) \\ & \searrow && \swarrow_{\nu} \\ && X }

which converges to some xXx \in X.

By the assumption that iIC i=\underset{i \in I}{\cap} C_i = \emptyset, there would exist an i xIi_x \in I such that xC i xx \neq C_{i_x}, and because C iC_i is a closed subset, there would exist even an open neighbourhood U xU_x of xx such that U xC i x=U_x \cap C_{i_x} = \emptyset. This would imply that x JU xx_J \neq U_x for all J{i x}J \supset \{i_x\}.

Now since the function ff defining the subset is cofinal, there would exist b 1Bb_1 \in B such that {i x}f(b 1)\{i_x\} \subset f(b_1). Moreover, by the assumption that the subnet converges, there would also be b 2Bb_2 \in B such that ν b 2U x\nu_{b_2 \leq \bullet} \in U_x. Since BB is directed, there would then be an upper bound bb 1,b 2b \geq b_1, b_2 of these two elements. This hence satisfies both ν f(e)U x\nu_{f(e)} \in U_x as well as {i x}f(b 1)f(b)\{i_x\} \subset f(b_1) \subset f(b). But the latter of these two means that ν f(b)\nu_{f(b)} is not in U xU_x, which is a contradiction to the former. Thus we have a proof by contradiction.

Logic of nets

A property of elements of a set XX (given by the subset SXS \subset X of those elements of XX satisfying this property) may be applied to nets in XX.

Being eventually in SS, def. 3, is a weakening of being always in SS (given by the universal quantifier ν\forall_\nu), while being frequently in SS is a strengthening of being sometime in SS (given by the particular quantifier ν\exists_\nu). Indeed we can build a formal logic out of these. Use essi,p[ν i]\ess\forall i, p[\nu_i] or ess νp\ess\forall_\nu p to mean that a predicate pp in XX is eventually true, and use essi,p[ν i]\ess\exists i, p[\nu_i] or ess νp\ess\exists_\nu p to mean that pp is frequently true. Then we have:

νpess νpess νp νp\forall_\nu p \;\Rightarrow\; \ess\forall_\nu p \;\Rightarrow\; \ess\exists_\nu p \;\Rightarrow\; \exists_\nu p
ess ν(pq)ess νpess νq\ess\forall_\nu (p \wedge q) \;\Leftrightarrow\; \ess\forall_\nu p \wedge \ess\forall_\nu q
ess ν(pq)ess νpess νq\ess\exists_\nu (p \wedge q) \;\Rightarrow\; \ess\exists_\nu p \wedge \ess\exists_\nu q
ess ν(pq)ess νpess νq\ess\forall_\nu (p \vee q) \;\Leftarrow\; \ess\forall_\nu p \wedge \ess\forall_\nu q
ess ν(pq)ess νpess νq\ess\exists_\nu (p \vee q) \;\Leftrightarrow\; \ess\exists_\nu p \vee \ess\exists_\nu q
ess ν¬p¬ess νp\ess\forall_\nu \neg{p} \;\Leftrightarrow\; \neg\ess\exists_\nu p

and other analogues of theorems from predicate logic. Note that the last item listed requires excluded middle even though its analogue from ordinary predicate logic does not.

A similar logic is satisfied by ‘almost everywhere’ and its dual (‘not almost nowhere’ or ‘somewhere significant’) in measure spaces.

References

A textbook account is in

Lecture notes include

Revised on May 15, 2017 05:57:37 by Urs Schreiber (195.37.209.183)