Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31
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The punctured disc is connected but not simply connected

Statement refuted

Every connected subdomain of the unit disc is simply connected.

Facts & Assumptions

Given: The punctured disc Ω={z∈C:0<∣z∣<1} and the circle γ(t)=12eit on [0,2π].

[L1]

In the grand theorem, connected spherical complement is equivalent to simple connectivity (For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent).

[L2]

A circle around the origin has winding number 1 about the origin (A circle traversed k times has winding number k inside and 0 outside).

Counterexample

technique · direct
1.1

The punctured disc is open and connected, and the loop γ lies in it. Its spherical complement is the union of the singleton {0} and the outer region {∣z∣≥1}∪{∞}, so it is disconnected.

given
2.1

By [L1], step 1.1 shows that the punctured disc is not simply connected. Fact [L2] gives the same geometric reason numerically: the loop γ still winds once around the omitted point 0.

step 1.1L1L2
3.1

This refutes the claim.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources