EDB — 0KD

[0KD]Prerequisites:[0KC]↺↻.Let \(X,Y\) be topological Hausdorff space. Let \(E⊆ X\), let \(f:E→ Y\), and suppose that \(x_ 0\) is an accumulation point of \(E\) in \(X\).

• If \(\lim _{x→ x_ 0}f(x)=ℓ\) then, for each net \(φ:J→ X\) with \(\lim _{j∈ J} φ(j) = x_ 0\) we have \(\lim _{j∈ J} f(φ(j)) = ℓ\).

• Consider the filtering set \(J\) given by the neighborhoods of \(x_ 0\); ¹ consider nets \(φ:J→ X\) with the property that \(φ(U)∈ U⧵\{ x_ 0\} \) for each \(U∈ J\). We note that \(\lim _{j∈ J} φ(j) = x_ 0\).

  If for each such net \(\lim _{j∈ J} f(φ(j)) = ℓ\), then \(\lim _{x→ x_ 0}f(x)=ℓ\).