8.6 Nets[2B6]

We will use the concepts of direct order, filtering order and cofinal set already discussed in Sec. ??. In the following \((Y, σ)\) will be a Hausdorff topological space.

Definition 261

[0K4] Let \((Y, σ)\) be a Hausdorff topological space. Let \((J,≤)\) be a set with filtering order (defined in 53). Let \(φ:J→ Y\) be a net (already met in Sec. 7.4).

We define that \(\lim _{j∈ J} φ(x) = ℓ ∈ Y\) if and only if, for every neighborhood \(V\) of \(ℓ\) in \(Y\) you have that \(φ(j)∈ V\) eventually for \(j∈ J\).

The definition of eventually is in 62, and it means that there exists \(k∈ J\) such that for every \(j≥ k\) you have \(φ(j)∈ V\).

The remark 236 holds in this case as well.

Definition 262

[2B4] Given a net \(x:J\to Y\), a point \(z∈ Y\) is said to be a limit point for \(x\) if there is a subnet \(y:H\to Y\) such that \(\lim _{j\in H} y(j)=z\).

(Note that “subnet” is intended in the general sense presented at the end of 238, where \(y=x\circ i\) by means of a map \(i:H\to J\) satisfying ??).

E262

[0K5] Prerequisites:3.Let \(J\) be a directed but non-filtering set; then let \(m∈ J\) be its maximum (which exists as seen in 3); if we define \(\lim _{j∈ J} φ(x)\) as in 261, show that the limit always exists and it is \(φ(m)\).

E262

[0K6]Let \((Y, σ)\) be a Hausdorff topological space and \(A⊆ Y\). Show that \(\overline A\) coincides with the set of all possible limits of nets \(φ:J→ A\) (varying \(J\) and then \(φ\)).

E262

[0K7]Let \((Y, σ)\) be a Hausdorff topological space and \(A⊆ Y\). Show that \(x∈ Y\) is an accumulation point for \(A\) if and only if there is a \(J\) filtering set and there is a net \(φ:J→ A⧵\{ x\} \) such that \(\lim _{j∈ J} φ(x) = x\).

E262

[2B7]Prerequisites:58,238.Difficulty:**.

Let \((Y,\sigma )\) be a Hausdorff topological space. Let \(J\) be a filtering set and \( x:J\to Y\) be a net in \(Y\). For every \(α ∈ J\) define \(E_{𝛼}{\stackrel{.}{=}}\{ x_{𝛽} : 𝛽 ∈ J, 𝛽 ≥ 𝛼 \} \) and

\[ E= ⋂ _{𝛼 ∈ J}\overline{E_{𝛼 }} \]

Prove that \(E\) coincides with the set \(L\) of limit points (defined in 262).

Hidden solution: [UNACCESSIBLE UUID ’2FK’]

E262

[0K8] Prerequisites:58,238,4.Difficulty:**.

Let \((Y,\sigma )\) be a Hausdorff topological space. Show that \(Y\) is compact if and only if every net taking values in \(Y\) admits a converging subnet.

Hidden solution: [UNACCESSIBLE UUID ’0K9’]