In a first-countable topological space, a function f: X → Y is continuous if and only if which of the following holds?
Af maps open sets to open sets
Bf maps every convergent sequence (xₙ → x) to a convergent sequence (f(xₙ) → f(x))
Cf is uniformly continuous on every compact subset of X
Df has a countable range
First-countability is precisely the condition that makes sequences sufficient to detect continuity. In an arbitrary topological space, you need nets or filters to characterize continuity — a function can be sequentially continuous without being truly continuous. But in a first-countable space, the countable neighborhood base at each point means that every convergence question can be answered by sequences alone. Sequential continuity and continuity coincide. This is the central power of first-countability.
Question 2 Multiple Choice
Which of the following topological spaces is NOT first-countable?
AThe real line ℝ with the standard (metric) topology
BAny finite topological space
CAn uncountable product ℝᴵ where I is an uncountable index set
DAny metric space with countably many points
The uncountable product ℝᴵ fails first-countability. At any point x, a neighborhood base must accommodate constraints on all possible finite coordinate subsets — but since I is uncountable, no countable collection of neighborhoods can probe every local constraint. In contrast, metric spaces (including ℝ) are always first-countable via the balls B(x, 1/n), and finite spaces have finite (hence countable) neighborhood bases. The uncountable product is the canonical example where sequences become genuinely insufficient.
Question 3 True / False
Every metric space is first-countable, because the open balls B(x, 1/n) for n = 1, 2, 3, … form a countable neighborhood base at each point x.
TTrue
FFalse
Answer: True
In any metric space, if U is any open set containing x, then by definition of openness there exists some ε > 0 with B(x, ε) ⊆ U. Choose n large enough that 1/n < ε; then B(x, 1/n) ⊆ U. So the collection {B(x, 1/n) : n ∈ ℕ} is a countable neighborhood base at x — every neighborhood of x contains some member of this collection. This confirms that all metric spaces satisfy first-countability, which is why metric-space intuition about sequences carries over to first-countable spaces generally.
Question 4 True / False
In any topological space, a point p is in the closure of a set A if and only if p is the limit of a sequence of points in A.
TTrue
FFalse
Answer: False
This characterization of closure via sequences holds in first-countable spaces but fails in general topological spaces. In a space that is not first-countable, a point can be in the closure of A without being the limit of any sequence from A — you need nets or filters to detect closure correctly. For example, in the uncountable product ℝᴵ, sequences are insufficient and nets are required. First-countability is exactly the condition that restores this sequential characterization of closure.
Question 5 Short Answer
Why does first-countability allow sequences to replace nets and filters in describing convergence, continuity, and closure?
Think about your answer, then reveal below.
Model answer: First-countability guarantees that each point x has a countable neighborhood base {Uₙ} — a countable collection of neighborhoods of x such that every neighborhood of x contains some Uₙ. This means that any topological fact about x (whether a function is continuous at x, whether x is a limit point of a set, whether a net converges to x) can be 'witnessed' by a sequence: you can always construct a sequence xₙ ∈ Uₙ that converges to x, and test every relevant property using that sequence. Without first-countability, no countable probe suffices — you need uncountably many 'directions' of approach simultaneously, which only nets or filters can capture.
The intuition is that a countable neighborhood base gives you enough resolution to detect the entire local structure at a point using sequences. Metric spaces are first-countable precisely because the balls B(x, 1/n) provide this resolution. When no countable base exists (as in uncountable products), sequences are blind to some topological features — a sequence might fail to converge to x even though every net-theoretic neighborhood of x is eventually entered. First-countability is the minimum condition to use 'metric-space thinking' about sequences in a purely topological setting.