Questions: Neighborhood Basis and Local Bases

Question 1

A neighborhood basis ℬ(x) at point x is given. A sequence (xₙ) converges to x if and only if:

• A. The sequence is eventually constant and equal to x
• B. For every basis element B ∈ ℬ(x), there exists N such that xₙ ∈ B for all n > N
• C. The sequence passes through every B ∈ ℬ(x) at least once
• D. The sequence is eventually inside every open set in the topology, which must be checked directly

Answer: (B)

By definition, xₙ → x means every open neighborhood of x eventually contains the tail of the sequence. Because every open neighborhood of x contains some B ∈ ℬ(x), it suffices to check only the basis elements — a much smaller collection. If the sequence is eventually in every basis element, it is eventually in every open neighborhood (since every open neighborhood contains a basis element). This is why a neighborhood basis 'characterizes' convergence at x: you only need to verify the basis sets, not all open sets.

Question 2

In a topological space with no countable neighborhood basis at some point x, which phenomenon can occur?

• A. Every open set containing x must be uncountable in cardinality
• B. Continuous functions at x cannot be defined in the usual way
• C. The point x can be a limit point of a set A even though no sequence from A converges to x
• D. Homeomorphisms cannot preserve the local topology at x

Answer: (C)

In spaces lacking a countable neighborhood basis (non-first-countable spaces), sequences are insufficient to detect all topological structure. A point x can be in the closure of A — meaning every neighborhood of x intersects A — yet no sequence from A converges to x, because sequences can only probe countably many neighborhoods. This failure is why sequences are replaced by nets or filters in general topology. It also explains why the standard calculus intuition ('take a sequence approaching x') breaks down outside metric spaces.

Question 3

In any metric space, the collection of open balls {B(x, 1/n) : n ∈ ℕ} forms a countable neighborhood basis at every point x.

• T. True
• F. False

Answer: True

For any open set U containing x in a metric space, there exists ε > 0 with B(x, ε) ⊆ U. Choose n large enough that 1/n < ε — then B(x, 1/n) ⊆ B(x, ε) ⊆ U. So every open neighborhood of x contains some ball B(x, 1/n). This countable collection therefore satisfies the neighborhood basis definition and is what makes metric spaces first-countable — and why ε–δ arguments using sequences work for all continuous-function questions in metric spaces.

Question 4

In nearly every topological space, a point x is in the closure of a set A if and mainly if some sequence of points from A converges to x.

• T. True
• F. False

Answer: False

This equivalence holds in first-countable spaces (including all metric spaces) but fails in general. In non-first-countable spaces, x can be a limit point of A — every open neighborhood of x intersects A — yet no sequence from A converges to x. The correct general statement uses nets or filters instead of sequences. This is a significant reason why working topologists cannot rely solely on sequential intuition and need the more general convergence notions.

Question 5

Why do sequences sometimes fail to characterize topology in general topological spaces, and what is the significance of having a countable neighborhood basis?

Model answer: Sequences can only probe countably many neighborhoods of a point. When a space has a countable neighborhood basis (first-countable), this is enough — every open neighborhood contains a basis element, so checking the countably many basis elements via a sequence is sufficient. In non-first-countable spaces, the local topology is richer than any sequence can fully explore, so points can be limit points of sets without any sequence from those sets converging to them. Having a countable local base is the property that makes sequential methods (ε–δ arguments, sequential compactness) fully adequate.

The key insight is that sequences are an inherently countable tool — they test one neighborhood per term. A countable neighborhood basis matches this perfectly. Without it, the topology contains 'too much' local structure for sequences to detect, motivating the more general tools of nets (indexed by directed sets) and filters (collections of subsets). First-countability is what makes the familiar analysis toolkit work in abstract spaces.