Let $D$ be the region consisting of all points in a circle of radius 4 but not in a square of side length 4 , both centered at the origin.

Is $D$ simply connected?

Choose 1 answer:
(A) Yes
(B) No

Question

Let $D$ be the region consisting of all points in a circle of radius 4 but not in a square of side length 4 , both centered at the origin.

Is $D$ simply connected?

Choose 1 answer:
(A) Yes
(B) No

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Define the term "simply connected" for a region in the plane. ‖ ‖ step 2 ⋮ A region is "simply connected" if every loop in the region can be continuously shrunk to a point without leaving the region. ‖ ‖ step 3 ⋮ Examine the region $D$ which is a circle of radius 4 excluding the points within a square of side length 4, both centered at the origin. ‖ ‖ step 4 ⋮ Identify that loops around the square cannot be shrunk to a point within $D$ because they would have to leave the region to avoid the square. ‖ ‖ step 5 ⋮ Conclude that region $D$ is not "simply connected" because it contains holes (the area inside the square) where loops cannot be contracted to a point. ‖ ∻Answer∻ ⚹ (B) No ⚹ ∻Key Concept∻ ⚹Simply Connected Regions⚹ ∻Explanation∻ ⚹A region is simply connected if every loop can be shrunk to a point within the region without crossing the region's boundary. The presence of the square hole in region $D$ prevents some loops from being shrunk to a point, making $D$ not simply connected.⚹◊What is the formula for the area of a circle with a radius of 4?⍭ Generate me a similar question◊

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