Consider the topologist's sine curve: the closure of the graph of sin(1/x) for x > 0, which includes the segment {0} × [−1, 1]. How many connected components does this space have?
A2 — the graph portion and the vertical segment are separate components
B1 — the entire set is a single connected component
CInfinitely many — each oscillation of the sine curve is its own component
D0 — it has no components since it is not path-connected
The topologist's sine curve is connected — the closure of any connected set is connected, and the graph {(x, sin(1/x)) : x > 0} is the continuous image of a connected set. Therefore the entire space is one connected component. The classic confusion (option A) is mistaking 'not path-connected' for 'not connected.' The space cannot be joined by a path across the accumulation segment, but it is still topologically connected — it cannot be split into two disjoint open sets. Path-connectedness is strictly stronger than connectedness.
Question 2 Multiple Choice
The space X = ℝ \ {0, 1} (the real line with two points removed). What are its connected components?
ATwo components: (−∞, 0) ∪ (0, 1) and (1, ∞)
BThree components: (−∞, 0), (0, 1), and (1, ∞)
COne component, since ℝ is connected and removal of finitely many points doesn't disconnect it
DInfinitely many, since every rational is a boundary point
Removing 0 and 1 from ℝ creates three maximal connected pieces: (−∞, 0), (0, 1), and (1, ∞). Each open interval is connected and cannot be merged with another without crossing a removed point. Option A is wrong because (−∞, 0) and (0, 1) are separated by the missing point 0 — they cannot be joined in X. Option C is wrong: removing even a single point from ℝ disconnects it (ℝ \ {0} has two components).
Question 3 True / False
Connected components of a topological space are generally open sets.
TTrue
FFalse
Answer: False
Connected components are always closed — the closure of a connected set is connected, so the closure of a component is still connected and contained in the component by maximality, meaning the component equals its own closure. However, components are not necessarily open. In the rational numbers ℚ with the subspace topology, every component is a single point — closed but not open. Components are open only when the space is locally connected, which is an additional assumption not required by the definition.
Question 4 True / False
If two points in a topological space cannot be connected by a continuous path, they should lie in different connected components.
TTrue
FFalse
Answer: False
The topologist's sine curve is the canonical counterexample. The point (0, 0) on the vertical accumulation segment and any point on the sine graph cannot be connected by a path — the curve oscillates infinitely fast near the y-axis, preventing a continuous path from crossing. Yet the entire space is connected, so all points lie in the same connected component. Path-connectedness implies connectedness, but not conversely; failing to be path-connected does not imply lying in different connected components.
Question 5 Short Answer
Why is the connected component of a point defined as the union of all connected subsets containing that point, rather than by requiring a path between points?
Think about your answer, then reveal below.
Model answer: Paths are not always available — a space can be connected (one component) even when no path joins certain point pairs. Defining components via paths yields path-components, which can be strictly finer than connected components. The definition via 'union of all connected subsets' is purely topological and does not require any notion of a continuous curve through the space.
The subtlety matters in spaces like the topologist's sine curve, where path-components and connected components diverge. In locally path-connected spaces (most familiar geometric objects), the two notions coincide. But in general topology — and especially in algebraic topology, where one studies spaces with exotic local behavior — the distinction is essential. The connected component is the maximal connected subset; the path-component is the maximal path-connected subset. The former is always closed; the latter need not be.