Questions: Convergence of Sequences in Topological Spaces

Question 1

In the cofinite topology on an infinite set X (open sets are sets with finite complement, plus ∅), what happens to sequences of distinct points?

• A. No sequences converge, because the topology is too coarse to pin down a limit
• B. Each sequence converges to at most one point, just as in a metric space
• C. Every sequence of distinct points converges to every point in X simultaneously
• D. Sequences converge only to accumulation points of the underlying set

Answer: (C)

In the cofinite topology, every open set containing any point x has a finite complement — so it contains all but finitely many elements of X. For a sequence (xₙ) of distinct points and any point x, every open neighborhood of x excludes only finitely many terms, so the sequence is eventually inside every such neighborhood. Therefore the sequence converges to x — and simultaneously to every other point. This dramatically illustrates why limits need not be unique in non-Hausdorff spaces.

Question 2

A topology student claims: 'If x is a limit point of the set S, then there must be a sequence from S converging to x.' In a general topological space, this claim is:

• A. Always true — limit points are defined precisely by sequences approaching them
• B. False in general — in non-first-countable spaces, a point can be a limit point of S without any sequence from S converging to it; nets or filters are needed to detect all limit points
• C. True in all Hausdorff spaces regardless of countability
• D. True only in spaces where every open set is also closed

Answer: (B)

This is the key failure of sequences in general topology. In metric spaces and first-countable spaces, sequences detect limit points because every point has a countable neighborhood base. But in non-first-countable spaces (such as uncountable products with the product topology), a point can accumulate in a set without any sequence from that set converging to it — there are too many open sets and too few sequences to probe them all. Nets (indexed by directed sets) and filters are rich enough to detect all limit points. This is why sequences cannot fully characterize topology in general.

Question 3

In a Hausdorff topological space, every convergent sequence has exactly one limit.

• T. True
• F. False

Answer: True

The Hausdorff condition requires that any two distinct points have disjoint open neighborhoods. If (xₙ) converged to both x and y with x ≠ y, take disjoint open sets U ∋ x and V ∋ y. The sequence must eventually be in U (by convergence to x) and eventually in V (by convergence to y). But U ∩ V = ∅, so the sequence cannot eventually be in both — contradiction. The Hausdorff axiom is precisely the condition that forces limit uniqueness.

Question 4

In a general topological space, knowing which sequences converge and to what limits is sufficient to largely determine the topology.

• T. True
• F. False

Answer: False

This holds only in first-countable spaces (where every point has a countable neighborhood base), which includes all metric spaces. In spaces like an uncountable product with the product topology, or a space with the cocountable topology on an uncountable set, different topologies can agree on all sequence convergences yet be genuinely distinct. To fully characterize topology in general, nets (indexed by arbitrary directed sets) or filters are required. Sequences are indexed by ℕ and simply cannot probe all the open sets in non-first-countable spaces.

Question 5

Why do sequences fail to fully characterize convergence in general topological spaces? What is needed instead, and why does this limitation not arise in metric spaces?

Model answer: Sequences are indexed by the natural numbers — a countable index set. In a metric space, every point has a countable neighborhood base (the balls of radius 1/n), so all topological information about a point can be detected by a countable approach. But in general topological spaces, a point may have uncountably many 'directions' of approach, none of which can be captured by any single sequence. Nets generalize sequences by allowing the index set to be any directed set, and filters provide an equivalent framework using collections of subsets. Both are rich enough to detect all limit points and fully characterize the topology. The limitation does not arise in metric spaces because first-countability guarantees that sequences suffice — the countable neighborhood base acts as a 'decoder' from topological structure to sequential behavior.

First-countable spaces are precisely those where sequential convergence and topological convergence coincide. Beyond that class, topology and sequence behavior come apart — a function can preserve all convergent sequences without being continuous.