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Math 344-1,2: Kursobjekt

Aaron Greicius

Section 1.22 Nets

Nets are a relatively straightforward generalization of a sequence \((x_n)_{n\in \Z_+}\) obtained by replacing the index set \(\Z_+\) with the more general notion of a directed set. Our main application of the theory of nets is (a) to give a characterization of compactness in terms of convergence and limit points of nets (1.22.11), and (b) to use this characterization to prove the Tychonoff theorem

Definition 1.22.1. Partial ordering axioms.

Let \(R\) denote a binary relation on the set \(I\text{.}\) We write \(iRj\) to assert that the relation \(R\) holds between elements \(i,j\in I\text{.}\) The relation \(R\) is a partial ordering if it satisfies the following three axioms.
  • Reflexive (PO1).
    For all \(i\in I\text{,}\) we have \(iRi\text{.}\)
  • Transitive (PO2).
    For all \(i,j,k\in I\text{,}\) if \(iRj\) and \(jRk\text{,}\) then \(iRk\text{.}\)
  • Antisymmetric (PO3).
    For all \(i,j\in I\text{,}\) if \(iRj\) and \(jRi\text{,}\) then \(i=j\text{.}\)
The relation \(R\) is a quasiordering (or preordering) if it satisfies axioms PO1 and PO2. When \(R\) is a partial ordering or a quasiordering, we will often write \(i\leq j\) for \(iRj\text{.}\)
A partially ordered set (POSET, for short) is a pair \((I,\leq)\text{,}\) where \(\leq\) is a partial ordering on \(I\text{.}\) Similarly,a quasiordered set is a pair \((I,\leq)\text{,}\) where \(\leq\) is a a quasiordering on \(I\text{.}\)

Example 1.22.2. Common examples.

  1. The usual “less than or equal to” relation \(x\leq y\) is a partial ordering on \(\R\text{.}\)
  2. Let \(X\) be a set, and let \(I=\mathcal{P}(X)\) be the power set of of \(X\text{.}\) The relation \(A\leq B \iff A\subseteq B\) defines a partial ordering on \(I\text{.}\)
  3. Let \(X\) be a set, and let \(I=\mathcal{P}(X)\) be the power set of of \(X\text{.}\) The relation \(A\leq B \iff A\supseteq B\) also defines a partial ordering on \(I\text{.}\)
  4. For an element \(x\) in a topological space \(X\) the set \(I\) of all open neighborhoods of \(x\) is a partially ordered set under reverse inclusion: i.e., \(U\leq V\iff U\supseteq V\text{.}\)
  5. Recall that an integer \(m\) divides an integer \(n\text{,}\) written \(m\mid n\text{,}\) if there is an integer \(q\) such that \(n=mq\text{.}\) The relation \(m\leq n\iff m\mid n\) defines a quasiordering on \(\Z\text{,}\) but not a partial ordering: \(2\mid -2\) and \(-2\mid 2\text{,}\) but \(2\ne -2\text{.}\) The same relation does define a partial ordering on \(\Z_+\text{.}\)

Definition 1.22.3. Directed set.

A directed set is a pair \((I,\leq)\text{,}\) where \(\leq\) is a quasiordering on \(I\) that satisfies the following axiom.
  • Directed (D).
    For all \(i,j\in I\text{,}\) there is a \(k\in I\) such that \(i\leq k\) and \(j\leq k\text{.}\)
A subset \(J\) of a directed set \(I\) is cofinal if for all \(i\in I\) there is a \(j\geq i\) such that \(j\in J\text{.}\)

Example 1.22.4. Common examples.

Each quasiordered (and/or partially ordered) set in Example 1.22.2 is easily seen to satisfy the further axiom (D), and is thus a directed set.

Definition 1.22.5. Nets and convergent nets.

Let \(X\) be a set. A net in \(X\) is a function \(f\colon I\rightarrow X\) where \(I\) is a directed set. In particular, a net is a tuple, and accordingly we may use the coordinate notation \(f=(x_i)_{i\in I}\text{,}\) where \(x_i=f(i)\text{.}\)
A net \(f\colon I\rightarrow X\) is eventually in a subset \(A\subseteq X\) if there is an element \(i_0\in I\) such that \(x_j\in A\) for all \(j\geq i_0\text{.}\)
Assume \(X\) is a topological space. A net \(f\colon I\rightarrow X\) converges to an element \(x\in X\text{,}\) denoted \(x_i\rightarrow x\text{,}\) if for all open sets \(U\) containing \(x\) the net \(f=(x_i)_{i\in I}\) is eventually in \(U\text{.}\)
A net \(f\colon I\rightarrow X\) is cofinally in a subset \(A\subseteq X\) if the set \(J=f^{-1}(A)\) is cofinal: equivalently, if for all \(i\in I\) there is a \(j\geq i\) such that \(x_j=f(j)\in A\text{.}\)
An element \(x\in X\) is a limit point of the net \(f\colon I\rightarrow X\) if \(f=(x_i)\) is cofinally in every open neighborhood of \(x\text{.}\)

Example 1.22.6. Riemann integral.

A careful examination of the Riemann integral of a function over an interval \([a,b]\) reveals that it is a statement about a convergent net. Indeed, let \(I\) be the set of all “pointed partitions” of \([a,b]\) into subintervals: i.e., an element of \(I\) is a pair \((\mathcal{P}, \boldx)\text{,}\) where \(\mathcal{P}\) is a subinterval partition
\begin{equation*} \mathcal{P}\colon a=x_0 < x_1< \dots < x_n=b \end{equation*}
and \(\boldx=(x_1^*, x_2^*,\dots, x_n^*)\) is a choice of “sample points” in each subinterval of \(\mathcal{P}\text{.}\) The relation \((\mathcal{P},\boldx)\leq (\mathcal{P}', \boldx')\iff \mathcal{P}' \text{ is a refinement of } \mathcal{P}\) is a quasiordering on \(I\) satisfying (D). To any function \(f\colon [a,b]\rightarrow \R\) we can associate the net \((R_i)_{i\in I}\text{,}\) where for each \(i=(\mathcal{P}, \boldx)\in I\) we define \(R_i\) to be the corresponding Riemann sum of \(f\text{.}\) The statement that the Riemann integral \(\int_a^b f\, dx\) exists and is equal to \(R\) is then equivalent to the statement that the net \((R_i)_{i\in I}\) converges to \(R\text{.}\)

Proof.

  1. First we show the forward implication. Assume \(x\in \overline{A}\text{.}\) Let \(I\) be the set of all pairs \(i=(U,a)\text{,}\) where \(U\) is an open neighborhoods of \(x\text{,}\) and \(a\in U\cap A\text{.}\) The set \(I\) is nonempty since \(x\in \overline{A}\text{.}\) For elements \(i=(U,a)\) and \(i'=(U',a')\) of \(I\) define \(i\leq i'\) if and only if \(U\supseteq U'\text{.}\) It is easy to see that \(I\) is a quasi-ordered and directed set with respect to this relation. Lastly, define \(f\colon I\rightarrow A\) as \(f((U,a))=a\text{.}\) The net \(f\) converges to \(x\text{.}\) Indeed, for any open set \(V\) containing \(x\text{,}\) there is an element \(a\in V\cap A\text{.}\) Setting \(i_0=(V,a)\text{,}\) if \(j=(U,a')\) satisfies \(j\geq i_0\text{,}\) then \(U\subseteq V\) and hence \(f(j)=f((U,a'))=a'\in V\text{,}\) as desired.
    For the reverse implication, assume \(x_i\rightarrow x\) for some net \((x_i)_{i\in I}\) in \(A\text{.}\) Given any open set \(U\) containing \(x\text{,}\) there is an \(i_0\in I\) such that \(x_j\in U\) for all \(j\geq i_0\text{.}\) In particular, we have \(x_{i_0}\in U\text{.}\) Since \((x_i)_{i\in I}\) is a net in \(A\text{,}\) we conclude that \(x_{i_0}\in\cap A\cap U\text{.}\)
  2. Assume \(f\) is continuous and \((x_i)_{i\in I}\) is a net satisfying \(x_i\rightarrow x\text{.}\) Given any open set \(U\) containing \(y\text{,}\) the net \((x_i)_{i\in I}\) is eventually in \(f^{-1}(U)\text{,}\) since this is an open set containing \(x\text{.}\) But then clearly \((f(x_i))_{i\in I}\) is eventually in \(f(f^{-1}(U))\subseteq U\text{.}\)
    Inversely, if \(f\) is not continuous, then there is an \(x\in X\) and open set \(U\) containing \(y=f(x)\) such that for all open sets \(V\) containing \(x\text{,}\) there is an element \(x'\in V\) such that \(f(x')\in f(V)-U\text{.}\) Let \(I\) be the set of all pairs \((V, z)\text{,}\) where \(V\) is an open neighborhood of \(x\) and \(f(z)\in f(V)-U\text{.}\) The set \(I\) is nonempty and is directed under the quasi-ordering \((V,z)\leq (V',z')\) if and only if \(V\supseteq V'\text{.}\) Define the net \(h\colon I\rightarrow X\) as \(h((V,z))=z\text{.}\)
    The net \(h\) converges to \(x\text{:}\) given any open neighborhood \(V\) of \(x\text{,}\) there is a \(z\in V\) such that \(i_0=(V,z)\) is an element of \(I\text{;}\) for all \(j=(V',z')\geq i_0=(V,z)\text{,}\) we have \(V\subseteq V'\) and hence \(h(j)=h((V',z'))=z'\in V'\subseteq V\text{.}\)
    On the other hand, the net \(f\circ h\text{,}\) defined as \(f\circ h((V,z))=f(z)\) does not converge to \(f(x)\text{.}\) Indeed by construction, for all \(j=(V,z)\in J\) we have \(f(z)\notin U\text{.}\)
  3. Assume \(X\) is Hausdorff and that the net \((x_i)_{i\in I}\) converges to \(x\in X\text{.}\) Given any \(y\ne x\text{,}\) pick disjoint open sets \(U, V\) containing \(x\) and \(y\text{,}\) respectively. I claim that \((x_i)_{i\in I}\) is not eventually in \(V\text{.}\) Indeed, since \(x_i\rightarrow x\text{,}\) there is an \(i_0\in I\) such that \(x_i\in U\) for all \(i\geq i_0\text{.}\) If, similarly, there were a \(j_0\in I\) such that \(x_j\in V\) for all \(j\geq j_0\text{,}\) then we could find an index \(k\) satisfying \(k\geq i_0\) and \(k\geq j_0\text{,}\) in which case \(x_k\in U\cap V=\emptyset\text{.}\) Contradiction! We conclude that \((x_i)_{i\in I}\) is not eventually in \(V\text{,}\) and hence that \((x_i)_{i\in I}\) does not converge to \(y\text{.}\)
    Now assume that \(X\) is not Hausdorff, as witnessed by elements \(x\ne y\in X\text{:}\) this means for all open sets \(U, V\) containing \(x\) and \(y\text{,}\) respectively, we have have \(U\cap V\ne \emptyset\text{.}\) Let \(I\) be the set of all triples \((U,V, z)\text{,}\) where \(U, V\) are open neighborhoods of \(x\) and \(y\text{,}\) respectively, and \(z\in U\cap V\text{.}\) Declare \((U,V,z)\leq (U',V', z')\) if and only if \(U\supseteq U'\) and \(V\supseteq V'\text{.}\) The set \(I\) is directed: given \(i=(U, V, z_i), j=(U',V',z_j)\in I\text{,}\) we can take \(k=(U\cap U, V\cap V', z_k)\text{,}\) for some \(z_k\in U\cap U'\cap V\cap V'\text{.}\) Now, let \(f=(z_i)_{i\in I}\) be the net in \(X\) defined as follows: given \(i=(U, V,z)\text{,}\) define \(f(i)=z\text{.}\) I claim that \(z_i\rightarrow x\) and \(z_i\rightarrow y\text{.}\) Indeed given any open set \(U\) containing \(x\) and open set \(V\) containing \(y\text{,}\) there is an element \(z\in U\cap V\text{.}\) Let \(i_0=(U,V,z)\in I\text{.}\) For any \(i'=(U',V', z')\geq i_0\) we have \(f(i')=z'\in U'\cap V'\subseteq U\cap V\text{.}\) Thus \(f=(z_i)_{i\in I}\) is eventually in \(U\) and eventually in \(V\text{,}\) as desired.

Definition 1.22.8. Subnets.

Let \((J,\leq_J)\) and \((I,\leq_I)\) be quasiordered sets. An order-preserving map is a function \(g\colon J\rightarrow I\) satisfying the following property: if \(j\leq_J j'\text{,}\) then \(g(j)\leq_I g(j')\text{.}\)
Let \(f=(x_i)_{i\in I}\) be a net in \(X\text{.}\) A subnet of \(I\) is a net of the form \(h=f\circ g\colon J\rightarrow X\text{,}\) where \(J\) is a directed set, and \(g\colon J\rightarrow I\) is an order-preserving map, and the image \(g(J)\) is cofinal in \(I\text{.}\) Using tuple notation, we will write \(h=(x_{g(j)})_{j\in J}\) for the subnet \(h=f\circ g\text{.}\)

Proof.

Let \(x\) be a limit point of the net \((x_i)_{i\in I}\text{.}\) By definition of limit point, the collection \(\mathcal{A}\) of all open neighborhoods of \(x\) satisfies condition (i) of Kelley’s lemma; condition (ii) follows from properties of open sets. We conclude that there is a subnet which is eventually in \(U\) for all open sets containing \(x\text{,}\) and thus that this subnet converges to \(x\text{.}\)
Assume \(x\) is not a limit point of \(f=(x_i)_{i\in I}\text{.}\) Let \(U\) be an open neighborhood of \(x\) such that \(I_U=\{i\in I\colon x_i\in U\}\) is not cofinal; then there exists an \(i_0\in I\) such that \(x_i\notin U\)for all \(i\in I_U\) with \(i\geq i_0\text{.}\) Let \(h=f\circ g\colon J\rightarrow X\) be a subnet of \(f\text{.}\) We can write \(h=(x_{g(j)})_{j\in J}\text{.}\) I claim that \(x_{g(j)}\not\to x\text{.}\) Indeed, since \(g(J)\) is cofinal in \(I\) we can find a \(j_0\) such that \(g(j_0)\geq i_0\text{,}\) in which case for all all \(j\geq j_0\) we have \(g(j)\geq g(j_0)\geq i_0\text{,}\) and thus \(x_{g(j)}\notin U\text{.}\) This shows that in fact \((x_{g(j)})_{j\in J}\) is eventually in \(X-U\text{;}\) in particular it is definitely not eventually in \(U\text{.}\)

Proof.

Assume \(\mathcal{A}\) is a collection of subsets of \(X\) satisfying conditions (i)-(ii). Define \(J\) to be the set of pairs \((i, A)\) where \(A\in \mathcal{A}\) and \(x_i\in A\text{.}\) Given \(j=(i, A)\text{,}\) \(j'=(i', A')\text{,}\) we declare \(j\leq j'\) if and only if \(i\leq i'\) and \(A\supseteq A'\text{.}\) It is clear that this is a quasiordering. We now show \(J\) is directed. Given \(j=(i,A)\) and \(j'=(i', A')\text{,}\) by property (ii) there is an \(A''\in \mathcal{A}\) such that \(A''\subseteq A\cap A'\text{.}\) Since \(f^{-1}(A'')\) is cofinal, there is an \(i''\in I\) such that \(i''\in A''\text{,}\) \(i''\geq i\) and \(i''\geq i'\text{.}\) Setting \(j''=(i'', A'')\text{,}\) we see that \(j\leq j''\) and \(j'\leq j''\text{.}\)
Define \(g\colon J\rightarrow I\) as \(g((i,A))=i\text{.}\) It is clear that \(g\) is order preserving. We show that \(g(J)\) is cofinal in \(I\text{.}\) Given any \(i\in I\text{,}\) pick any \(A\in \mathcal{A}\text{.}\) Since \(f^{-1}(A)\) is cofinal, there is an \(i'\geq i\) such that \(i'\in f^{-1}(A)\text{.}\) We have \(i'=g((i', A))\text{.}\)
Lastly, we show that the subnet \((x_{g(j)})_{j\in J}\) is eventually in \(A\) for all \(A\in \mathcal{A}\text{.}\) Indeed, given \(A\in \mathcal{A}\text{,}\) let \(j_0=(i, A)\) for any \(i\in f^{-1}(A)\text{.}\) If \(j\geq j_0\) for \(j=(i', A')\text{,}\) then we have \(i\leq i'\) and \(A'\subseteq A\text{.}\) It follows that \(x_{g(j)}=i'\in A'\subseteq A\text{.}\) This proves \(x_{g(j)}\in A\) for all \(j\geq j_0\text{,}\) and thus that \((x_{g(j)})_{j\in J}\) is eventually in \(A\text{.}\)

Proof.

Statements (ii) and (iii) are immediately seen to be equivalent thanks to Theorem 1.22.9. We will show that (i) and (ii) are equivalent.

Implication: (i) \(\implies\) (ii).

Let \(f=(x_i)_{i\in I}\) be a net in \(X\text{.}\) For each \(i\in I\) define \(C_i=\overline{\{x_j\colon j\geq i\}}\text{.}\) Clearly each \(C_i\) is closed and nonempty. Furthermore, for any finite collection \(i_1,i_2,\dots, i_n\) we have \(\bigcap_{k=1}^n C_{i_k}\ne \emptyset\text{,}\) since we can find a \(j\) such that \(j\geq i_k\) for all \(1\leq k\leq n\text{,}\) and hence \(x_j\in \bigcap_{k=1}^n C_{i_k}\text{.}\) Since \(X\) is assumed to be compact, there is an element \(x\in \bigcap_{i\in I}C_i\text{.}\) We show that \(x\) is a limit point of \((x_i)_{i\in I}\text{.}\) Indeed, given any open set \(U\) containing \(x\) and any \(i\in I\text{,}\) since \(x\in C_i=\overline{\{x_j\colon j\geq i\}}\text{,}\) there is an element \(x_j\) with \(j\geq i\) such that \(x_j\in U\text{.}\) This proves that \((x_i)_{i\in I}\) is cofinally in \(U\text{,}\) as desired.

Implication: (ii) \(\implies\) (i).

We prove the contrapositive. Suppose \(X\) is not compact, and let \(\mathcal{U}=\{U_i\}_{i\in I}\) be a cover with no finite cover. Define \(J\) to the be the set of all pairs \((K, x)\text{,}\) where \(K\) is a finite subset of \(I\) and \(x\in X-\bigcup_{k\in K}U_k\text{.}\) (Note that \(X-\bigcup_{k\in K}U_k\ne\emptyset\) since \(\mathcal{U}\) has no finite subcover.) It is easy to see that \(J\) is a directed set under the quasiordering defined as \((K, x)\leq (K', x')\) if and only if \(K\subseteq K'\text{.}\) Define the net \(f\colon J\rightarrow X\) as \(f((K, x))=x\text{.}\) We show that \(f\) has no limit point. To this end, given any \(y\in Y\text{,}\) we have \(y\in U_{i_0}\) for some \(i_0\in I \text{.}\) We will show that the net \(f\) is not cofinally in \(U_{i_0}\text{.}\) To this end consider the index \(j=({i_0}, x)\in J\text{,}\) where \(x\in X-U_{i_0}\text{.}\) For all \(j'=(K, x)\) with \(j'\geq j\text{,}\) we have \(K\supseteq \{i_0\}\text{,}\) in which case \(f(j')=x\in X-\bigcup_{k\in K}U_k\subseteq X-U_{i_0}\text{.}\) Thus there is no \(j'\geq j\) satisfying \(f(j)\in U_{i_0}\text{.}\)
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