Is the closed interval [0.9, 1.1] a neighborhood of the point 1 in ℝ with the standard topology?
ANo — a neighborhood must itself be an open set, and [0.9, 1.1] is closed
BYes — [0.9, 1.1] contains the open interval (0.95, 1.05) which contains 1
CNo — only open balls qualify as neighborhoods in metric spaces
DYes — but only after redefining the topology to include closed sets
A neighborhood of x is any set N containing an open set U with x ∈ U ⊆ N. The closed interval [0.9, 1.1] is a neighborhood of 1 because it contains the open interval (0.95, 1.05), which is open and contains 1. Neighborhoods do not need to be open — that is the point of the definition. Option A expresses the most common misconception: that 'neighborhood' and 'open set' are synonyms. The whole power of the neighborhood concept is precisely that it allows non-open sets to capture local structure.
Question 2 Multiple Choice
The open balls {B(x, 1/n) : n ∈ ℕ} form a neighborhood base at x in any metric space. What property of the space does this imply?
AThe space is second-countable: a single countable base covers the entire topology
BThe space is first-countable: every point has a countable neighborhood base
CSequences are insufficient to detect closure, so nets or filters are required
DEvery neighborhood of x can be expressed as a finite union of these balls
A countable neighborhood base at every point is exactly the definition of a first-countable space. All metric spaces are first-countable because the balls B(x, 1/n) form such a base: any neighborhood of x must contain some ball of this form. First-countability matters because it ensures sequences are sufficient to detect topological structure — x is in the closure of A iff some sequence in A converges to x. Option A confuses first-countability (countable base at each point) with second-countability (one countable base for the whole topology, a strictly stronger condition).
Question 3 True / False
A neighborhood of a point x should itself be an open set.
TTrue
FFalse
Answer: False
This is the central misconception about neighborhoods. A neighborhood of x is any set N that contains an open set U with x ∈ U ⊆ N — N itself need not be open. Closed intervals, half-open sets, or any set that 'wraps around' an open set containing x qualify. For example, [0, 1] is a neighborhood of 0.5 because it contains the open interval (0.3, 0.7). The definition is deliberately flexible: neighborhoods absorb boundary behavior while preserving the local information encoded in the open sets they contain.
Question 4 True / False
A set U is open in a topological space if and only if U is a neighborhood of each of its points.
TTrue
FFalse
Answer: True
This is a theorem connecting global and local perspectives on openness. If U is open, then U itself witnesses that U is a neighborhood of every x ∈ U (since U is open and x ∈ U ⊆ U). Conversely, if every point x ∈ U has some open set Uₓ with x ∈ Uₓ ⊆ U, then U = ∪{Uₓ : x ∈ U} is a union of open sets, hence open. This local characterization is useful because it lets you prove openness point-by-point rather than checking a global condition all at once.
Question 5 Short Answer
Why does first-countability matter for the relationship between sequences and topological structure?
Think about your answer, then reveal below.
Model answer: In a first-countable space, sequences are sufficient to capture all closure information: a point x is in the closure of a set A if and only if some sequence in A converges to x. The countable neighborhood base allows you to construct a sequence by picking one point from each ball B(x, 1/n) ∩ A. In spaces that fail first-countability, this construction breaks down — there can be points in the closure of A that no sequence in A approaches. In those spaces, the more general tools of nets or filters are required to detect topological structure.
The connection between sequences and topology is one reason metric spaces behave so much more tractably than general topological spaces: every metric space is first-countable, so the intuitions built up in real analysis (where sequences do all the work) transfer directly. Failure of first-countability is the precise obstruction that forces analysts working in function spaces or spaces of distributions to abandon sequences in favor of nets.