Questions: Locally Connected Spaces

Question 1

Which of the following is connected but NOT locally connected?

• A. The real line ℝ with the standard topology
• B. The topologist's sine curve (the closure of {(x, sin(1/x)) : x > 0})
• C. A disjoint union of two open intervals (0,1) ∪ (2,3)
• D. A circle S¹

Answer: (B)

The topologist's sine curve is connected — you cannot split it into two disjoint nonempty open sets — but it fails local connectedness near the origin. Every small neighborhood of the origin contains infinitely many disconnected arcs of the oscillating sine wave, so no connected open neighborhood of the origin exists. The real line ℝ and the circle S¹ are both connected AND locally connected. The disjoint union (0,1) ∪ (2,3) is locally connected but globally disconnected — the opposite failure.

Question 2

In a locally connected topological space, what can be said about the connected components?

• A. They are always closed but not necessarily open
• B. They are always open sets
• C. They are always finite in number
• D. They are always both open and closed only if the space is compact

Answer: (B)

Local connectedness implies that connected components are open. At any point x in a component C, local connectedness provides a connected open neighborhood U of x. Since U is connected and intersects C, U must be entirely contained in C. So every point of C has an open neighborhood inside C, making C open. In an arbitrary topological space, components are closed but not necessarily open — the rationals ℚ, for instance, have components that are single points, which are not open. Local connectedness is exactly the extra condition that forces components to be open.

Question 3

Most connected topological space is also locally connected.

• T. True
• F. False

Answer: False

The topologist's sine curve is the canonical counterexample: it is connected (the global space cannot be separated into two disjoint open sets) but not locally connected (near the origin, no small connected open neighborhood exists). Connectedness is a global property — it depends on the whole space. Local connectedness is a local property — it must hold at every point in small neighborhoods. A space can satisfy one without the other.

Question 4

A locally connected space can have more than one connected component.

• T. True
• F. False

Answer: True

Local connectedness and global connectedness are independent properties. The disjoint union (0,1) ∪ (2,3) is locally connected — every point has an obvious connected open neighborhood within its interval — but has exactly two connected components. Local connectedness says each point looks connected nearby; it says nothing about whether the whole space is connected. In locally connected spaces, those multiple components will each be open sets.

Question 5

What is the key structural consequence of a space being locally connected, and why does this consequence fail for arbitrary topological spaces?

Model answer: In a locally connected space, connected components are open sets. This holds because local connectedness provides a connected open neighborhood at every point, forcing each component to be open. In an arbitrary topological space, components are always closed but can fail to be open — in ℚ for example, components are single points, which are not open in the standard topology.

The openness of components is what makes locally connected spaces well-behaved for algebraic topology. It ensures clean partitions of the space, that path-lifting arguments work, and that universal covers exist. The failure in ℚ illustrates why: components so small they contain no open sets make local-to-global arguments impossible. Local connectedness is thus precisely the condition needed to prevent this pathological behavior.