A Note of Nets

Supplementary Exercises: Nets

We have already seen that sequences are “adequate” to detect limit points, continuous functions, and compact sets in metrizable spaces. There is a generalization of the notion of sequence, called a net, that will do the same thing for an arbitrary topological space. We give the relevant definitions here, and leave the proofs as exercises. Recall that a relation $\preceq$ on a set $A$ is called a partial order relation if the following conditions hold:

1. $\alpha \preceq \alpha$ for all $\alpha$.

2. If $\alpha \preceq \beta$ and $\beta \preceq \alpha$, then $\alpha = \beta$.

3. If $\alpha \preceq \beta$ and $\beta \preceq \gamma$, then $\alpha \preceq \gamma$.

Now we make the following definition:

A directed set $J$ is a set with a partial order $\preceq$ such that for each pair $\alpha, \beta$ of elements of $J$, there exists an element $\gamma$ of $J$ having the property that $\alpha \preceq \gamma$ and $\beta \preceq \gamma$.

1. Show that the following are directed sets:

   1. Any simply ordered set, under the relation $\preceq$.

   2. The collection of all subsets of a set $S$, partially ordered by inclusion (that is, $A \preceq B$ if $A \subseteq B$).

   3. A collection $\mathcal{A}$ of subsets of $S$ that is closed under finite intersections, partially ordered by reverse inclusion (that is $A \preceq B$ if $A \supseteq B$).

   4. The collection of all closed subsets of a space $X$, partially ordered by inclusion.

2. A subset $K$ of $J$ is said to be cofinal in $J$ if for each $\alpha \in J$, there exists $\beta \in K$ such that $\alpha \preceq \beta$. Show that if $J$ is a directed set and $K$ is cofinal in $J$, then $K$ is a directed set.

3. Let $X$ be a topological space. A net in $X$ is a function $f$ from a directed set $J$ into $X$. If $\alpha \in J$, we usually denote $f(\alpha)$ by $x_{\alpha}$. We denote the net $f$ itself by the symbol $(x_{\alpha})_{\alpha \in J}$, or merely by $(x_{\alpha})$ if the index set is understood.

   The net $(x_{\alpha})$ is said to converge to the point $x$ of $X$ (written $x_{\alpha} \to x$) if for each neighborhood $U$ of $x$, there exists $\alpha \in J$ such that Show that these definitions are reduced to familiar ones when $J = \mathbb{Z}_+$.

4. Suppose that $$(x_{\alpha}){\alpha \in J} \to x \text{ in } X \quad \text{and} \quad (y{\alpha})_{\alpha \in J} \to y \text{ in } Y.$$ Show that $(x_{\alpha}, y_{\alpha}) \to (x, y)$ in $X \times Y$.

5. Show that if $X$ is Hausdorff, a net in $X$ converges to at most one point.

6. Theorem. Let $A \subseteq X$. Then $x \in \overline{A}$ if and only if there is a net of points of $A$ converging to $x$.

   Hint: To prove the implication $\implies$, take as an index set the collection of all neighborhoods of $x$, partially ordered by reverse inclusion.

 

7. Theorem. Let $f : X \to Y$. Then $f$ is continuous if and only if for every convergent net $(x_{\alpha})$ in $X$, converging to $x$, say, the net $(f(x_{\alpha}))$ converges to $f(x)$.

8. Let $f : J \to X$ be a net in $X$; let $f(\alpha) = x_{\alpha}$. If $K$ is a directed set and $g : K \to J$ is a function such that

   1. $i \leq j \implies g(i) \preceq g(j)$,

   2. $g(K)$ is cofinal in $J$,

   then the composite function $f \circ g : K \to X$ is called a subnet of $(x_{\alpha})$. Show that if the net $(x_{\alpha})$ converges to $x$, so does any subnet.

9. Let $(x_{\alpha})_{\alpha \in J}$ be a net in $X$. We say that $x$ is an accumulation point of the net $(x_{\alpha})$ if for each neighborhood $U$ of $x$, the set of those $\alpha$ for which $x_{\alpha} \in U$ is cofinal in $J$. Lemma. The net $(x_{\alpha})$ has the point $x$ as an accumulation point if and only if some subnet of $(x_{\alpha})$ converges to $x$.

   Hint: To prove the implication $\implies$, let $K$ be the set of all pairs $(\alpha, U)$ where $\alpha \in J$ and $U$ is a neighborhood of $x$ containing $x_{\alpha}$. Define $(\alpha, U) \preceq (\beta, V)$ if $\alpha \preceq \beta$ and $V \subseteq U$. Show that $K$ is a directed set and uses it to define the subnet.

10. Theorem. $X$ is compact if and only if every net in $X$ has a convergent subnet.

    Hint: To prove the implication $\implies$, let $B_{\alpha} = \{x_{\beta} \mid \alpha \preceq \beta\}$ and show that $\{B_{\alpha}\}$ has the finite intersection property. To prove $\impliedby$, let $\mathcal{A}$ be a collection of closed sets having the finite intersection property, and let $\mathcal{B}$ be the collection of all finite intersections of elements of $\mathcal{A}$, partially ordered by reverse inclusion.

11. Corollary. Let $G$ be a topological group; let $A$ and $B$ be subsets of $G$. If $A$ is closed in $G$ and $B$ is compact, then $A \cdot B$ is closed in $G$.

    Hint: First give a proof using sequences, assuming that $G$ is metrizable.

12. Check that the preceding exercises remain correct if condition (2) is omitted from the definition of directed set. Many mathematicians use the term “directed set” in this more general sense.