A set X = {a, b, c} is given the indiscrete topology (only ∅ and X are open). The constant sequence a, a, a, ... converges to:
AOnly the point a, since the sequence is eventually constant at a
BThe points a and b but not c, since c was never a term of the sequence
CNo point, since the indiscrete topology lacks enough open sets to define convergence
DEvery point in X — all of a, b, and c simultaneously
A sequence converges to x if every open neighborhood of x contains all but finitely many terms. In the indiscrete topology, the only open set containing any point is all of X. Since X trivially contains every term of any sequence, every point qualifies as a limit. This is not a defect of the definition — it is a consequence of the topology being too coarse to separate points. Limit non-uniqueness is not a failure of convergence; it is a property of the topology itself.
Question 2 Multiple Choice
In a topological space that is NOT first-countable, why are sequences sometimes insufficient to describe the topology?
ASequences can only converge in metric spaces where distances are defined
BA point x may be a limit point of a set A — every open neighborhood of x meets A — yet no sequence from A converges to x, because sequences indexed by ℕ cannot probe all open neighborhoods when no countable neighborhood basis exists
CIn non-first-countable spaces, sequences have no well-defined ordering and cannot converge
DSequences only fail in finite topological spaces, not in infinite ones
Sequences are indexed by ℕ, so they can only 'probe' a limit point using countably many terms. If a point x has uncountably many incomparable open neighborhoods, a sequence may not eventually enter all of them — some neighborhoods are missed entirely. In a first-countable space, a countable neighborhood basis exists, so a sequence can detect all limit points by working through the basis. When first-countability fails, you need nets (indexed by general directed sets) to capture all topological information. This is one of the deepest differences between metric and general topological spaces.
Question 3 True / False
In a Hausdorff (T₂) topological space, every convergent sequence has a unique limit.
TTrue
FFalse
Answer: True
In a Hausdorff space, any two distinct points x and y have disjoint open neighborhoods U and V. If a sequence converged to both x and y, then U would need to contain all but finitely many terms AND V would also need to contain all but finitely many terms. But U and V are disjoint — no term can be in both — so this is impossible. The Hausdorff property is precisely what gives the topology enough 'separation' to force limits to be unique. Without it (as in the indiscrete topology), distinct points share all the same neighborhoods and can both be limits of any sequence.
Question 4 True / False
In a general topological space, a point x is in the closure of a set A if and only if some sequence of points from A converges to x.
TTrue
FFalse
Answer: False
This characterization holds in metric spaces and more generally in first-countable spaces, but fails in general topology. A point x can be a limit point of A — every open neighborhood of x meets A — without any sequence from A converging to x. This happens precisely in spaces that are not first-countable, where the natural numbers are not rich enough to index the neighborhoods around x. The correct generalization requires nets: x is in the closure of A if and only if some net from A converges to x. This is a core reason why nets, not sequences, are the fundamental notion of convergence in general topology.
Question 5 Short Answer
What property of a topological space guarantees that limits of sequences are unique, and why does that property force uniqueness?
Think about your answer, then reveal below.
Model answer: The Hausdorff (T₂) property guarantees unique limits. A space is Hausdorff if any two distinct points can be separated by disjoint open neighborhoods. If a sequence converged to two distinct points x and y, their separating neighborhoods U and V would each need to contain all but finitely many terms — but disjoint sets cannot both do this simultaneously, giving a contradiction.
The proof is short but structurally revealing: assume xₙ → x and xₙ → y with x ≠ y. By Hausdorff, find disjoint open U ∋ x and V ∋ y. By convergence to x, all but finitely many terms lie in U; by convergence to y, all but finitely many terms lie in V. But U ∩ V = ∅, so no term can be in both — a contradiction. Uniqueness of limits is thus a consequence of separation, not of convergence itself. This is why studying separation axioms (T₁, T₂, T₃, ...) matters: they control which convergence-like properties the topology supports.