Prof. Girardi Nets

We have already seen that sequences are ”adequate” to detect limit points, continuous functions, and compact sets in metric 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  on a set A is called a partial order relation if the following conditions hold: (1) α  α for all α (reflexive); (2) If α  β and β  α, then α = β (anti-symmetric); (3) If α  β and β  γ, then α  γ (transitive). A directed set J is a set with a partial order  such that for each pair α, β of elements of J, there exists an element γ of J having the property that α  γ and β  γ. Example 1. Show that the following are directed sets: (a) Any simply ordered set, under the relation ≤. (b) The collection of all subsets of a set S, partially ordered by inclusion (that is, A  B if A ⊆ B). (c) A collection A of subsets of S that is closed under finite intersections, partially ordered by reverse inclusion (that is, A  B if A ⊇ B). (d) The collection of all closed subsets of a space X, partially ordered by inclusion. A subset K of J is said to be cofinal in J if for each α ∈ J, there exists β ∈ K such that α  β. Show that if J is a directed set and K is cofinal in J, then K is a directed set. Definition 2. Let X be a topological space. A net in X is a function f from a directed set J into X. If α ∈ J, we usually denote f(α) by xα. We denote the net f itself by the symbol (xα)α∈J , or merely by (xα) if the index set is understood. The net (xα) is said to converge to the point x of X (written xα → x) if for each neighborhood U of x, there exists α ∈ J such that

α  β =⇒ xβ ∈ U. Show that these definitions reduce to familiar ones when J = N. Remark 3. (1) Suppose that

(xα)α∈J −→ x in X and (yα)α∈J −→ y in Y. Show that (xα × yα) −→ x × y in X × Y . (2) Show that if X is Hausdorff, a net in X converges to at most one point. Theorem 4. Let A ⊆ X. Then x ∈ A if and only if there is a net of points of A converging to x. [Hint: To prove the implication ⇒, take as index set the collection of all neighborhoods of x, partially ordered by reverse inclusion.] Theorem 5. Let f : X → Y . Then f is continuous if and only if for every convergent net (xα) in X, converging to x, say, the net (f(xα)) converges to f(x).

Definition 6. Let f : J → X be a net in X; let f(α) = xα. If K is a directed set and g : K → J is a function such that (1) i  j ⇒ g(i)  g(j), (2) g(K) is cofinal in J, then the composite function f ◦ g : K → X is called a subnet of (xα).

Show that if the net (xα) converges to x, so does any subnet. Let (xα)α∈J be a net in X. We say that x is an accumulation point of the net (xα) if for each neighborhood U of x, the set of those α for which xα ∈ U is cofinal in J.

Lemma 7. The net (xα) has the point x as an accumulation point if and only if some subnet of (xα) converges to x. yr.mn.dy: 20.01.12 Page 1 of 2 Prof. Girardi Nets

[Hint: To prove the implication ⇒, let K be the set of all pairs (α, U) where α ∈ J and U is a neighborhood of x containing xα. Define (α, U)  (β, V ) if α  β and V ⊆ U. Show that K is a directed set and use it to define the subnet.] Theorem 8. X is compact if and only if every net in X has a convergent subnet.

[Hint: To prove the implication ⇒, let Bα = {xβ : α  β} and show that (Bα) satisfies the finite intersection condition. To prove ⇐, let A be a collection of closed sets satisfying the finite intersection condition, and let B be the collection of all finite intersection of elements of A, partially ordered by reverse inclusion.] Remark 9. 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. Recall 10. Let (X, T ) be a topological space and x ∈ X. (1) A family B ⊆ T is called a base provided any T ∈ T is the union of sets in B. (2) A neighborhood of x is a set U ⊆ X which contains an open set V containing x. (3) A neighborhood base at x is a subcollection Bx taken from the neighborhoods of x where Bx has the property that for each neighborhood U of x there is some V ∈ Bx which is contained in U. Definition 11. (1) A toplogical space X is called separable if and only if it has a count- able dense set. (2) A topological space X is called first countable if and only if each point x ∈ X has a countable neighborhood base. (3) A topological space X is called second countable if and only if X has a countable base. Proposition 12. (1) Every metric space is first countable. (2) A metric space is second countable if and only if it is separable. (3) Any second countable topological space is separable. Remark 13. If one replaces the word “net” by sequence in the above statements, then • Theorems 4 and 5 remain true if X is first countable, • Theorem 8 remains true if X is second countable.

References [Mun] James R. Munkres, Topology: a first course. Prentice-Hall, Englewood-Cliffs, NJ, 1975. [RS-I] Michael Reed and Barry Simon, Methods of modern mathematical physics I, Second edition. Academic Press Inc., New York, 1980.

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