Questions: Neighborhoods and Open Sets

Question 1

In a topological space X, a subset U is open if and only if:

• A. U contains at least one neighborhood of some point in U
• B. U is a neighborhood of every point it contains
• C. U has no boundary points inside it
• D. U contains a neighborhood of every point in its complement

Answer: (B)

The neighborhood characterization of open sets is the key equivalence: U is open if and only if for every x ∈ U, U itself is a neighborhood of x — that is, U is an open set containing x. This means every point of U has 'room to move around inside U.' Option A is too weak (a single point having a neighborhood in U doesn't make U open); the condition must hold at every point. This equivalence lets you translate between the global open-set perspective and the local neighborhood perspective.

Question 2

You want to verify that a map f: X → Y is continuous at x ∈ X using neighborhoods. You have shown that for every open set V containing f(x), the preimage f⁻¹(V) contains an open set around x. This is:

• A. Sufficient for continuity at x — this is exactly the neighborhood definition
• B. Not sufficient — you need f⁻¹(V) itself to equal an open set, not just contain one
• C. Sufficient only if X is a metric space where open sets are open balls
• D. Not sufficient — you need to verify the condition at every point in X, not just at x

Answer: (A)

A function f is continuous at x if and only if: for every neighborhood V of f(x), f⁻¹(V) is a neighborhood of x. Since f⁻¹(V) containing an open set around x means f⁻¹(V) is itself a neighborhood of x (neighborhoods are closed under supersets), this is exactly the correct condition. Option B is wrong: a neighborhood of x need not equal an open set — it just needs to contain one. The key insight is that continuity is a local condition expressible purely in terms of neighborhoods.

Question 3

In a topological space, a sequence (xₙ) converges to x if and only if every neighborhood of x contains all but finitely many terms of the sequence.

• T. True
• F. False

Answer: True

This is the precise topological definition of convergence in neighborhood language. For any open set U containing x, there must exist N such that xₙ ∈ U for all n > N — the sequence is eventually inside every neighborhood of x. This generalizes the metric-space definition (where neighborhoods are ε-balls) to arbitrary topological spaces, and illustrates why neighborhoods are the natural vocabulary for convergence: the question 'does the sequence converge to x?' reduces to 'does every neighborhood of x eventually trap the sequence?'

Question 4

Two topologies on a set X can assign exactly the same neighborhoods to nearly every point while still being distinct topologies.

• T. True
• F. False

Answer: False

This is false — the neighborhood structure completely determines the topology. A set U is open if and only if it is a neighborhood of every point it contains. So if two topologies agree on which sets are neighborhoods of each point, they must agree on which sets are open, meaning they are identical topologies. This is the deep content of the equivalence between the open-set axioms and the neighborhood-filter axioms: the local neighborhood data at each point encodes the entire global topology.

Question 5

Why is the shift from thinking about 'open sets of the whole space' to 'neighborhoods of individual points' conceptually significant in topology?

Model answer: The shift matters because the properties we care about — continuity, convergence, limit points — are inherently local. They depend on what happens near a specific point, not on the global structure of the space. Neighborhoods focus attention exactly where needed: instead of asking 'is this set open globally?' we ask 'does this set contain a region around x?' at each point. This decomposition reveals that continuity, convergence, and limit points all reduce to questions about which sets contain a given point, and that the whole topology is encoded in these local neighborhood structures.

The neighborhood perspective makes local analysis modular and compositional. Continuity becomes 'neighborhoods of x map to neighborhoods of f(x),' convergence becomes 'the sequence eventually lands in every neighborhood of x,' and limit points become 'every neighborhood of x meets the set A.' These local formulations are cleaner than global ones and generalize cleanly beyond metric spaces, which is why the neighborhood framework is the right level of abstraction for topology.