Alphabeta Math

Connectedness: Examples and Counterexamples

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✓ 7 results · all verified · 5 also independently AI-judged
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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies placed in the connectedness hierarchy

Example

The zigzag curve and its closure worked out: the components, the path components, and the points at which local connectedness fails

Example

ℚ as a subspace of ℝ: every component is a single point, no point is isolated, and the space is not locally connected anywhere

Counterexample

In the subspace of ℝ² made of the vertical unit segments over 1/(n+1) together with the two points (0,0) and (0,1), the component of (0,0) is a singleton while its quasicomponent is {(0,0), (0,1)}

Example

The long ray is connected and locally connected, every proper initial segment is order-convex and connected, and, assuming countable choice, no at most countable subset is cofinal

Counterexample

The comb space is path-connected and fails to be locally connected at every point of the limit tooth strictly above the base, so path-connectedness does not imply local connectedness

Counterexample

ℝ^ℕ in the box topology is disconnected, the bounded and the unbounded sequences forming a separation, although every factor is connected and the product topology is connected

Sources