Circle vs Line Segment

Topology · Axiom Academy

EXAMPLE Circle vs Line Segment Three worked problems proving topological distinctions between S¹, [0,1], and ℝ Problem 1: S¹ is not homeomorphic to [0,1] Prove that the circle S¹ is not homeomorphic to the closed interval [0,1] using the "remove one point" argument. If two spaces X and Y are homeomorphic, removing a point from each should yield homeomorphic spaces. We'll use this to show S¹ and [0,1] cannot be homeomorphic. Why is connectedness relevant to this proof? Problem 2: S¹ is not homeomorphic to ℝ Prove that the circle S¹ is not homeomorphic to the real line ℝ using compactness. A space is compact if every open cover has a finite subcover. Compactness is preserved by homeomorphisms, so if X ≅ Y, then X is compact ⟺ Y is compact. Which property distinguishes S¹ from ℝ in this proof? Problem 3: S¹ \ point ≅ ℝ via Stereographic Projection Prove that removing one point from S¹ gives a space homeomorphic to ℝ using stereographic projection. Project from the "north pole" N = (0,1) of S¹ onto the x-axis (which we identify with ℝ). Each point P ≠ N on S¹ corresponds to where the line NP intersects the x-axis. What makes stereographic projection a homeomorphism? Connectedness distinguishes S¹ from [0,1]: Removing a point from S¹ leaves it connected, but removing an interior point from [0,1] disconnects it. Compactness distinguishes S¹ from ℝ: S¹ is compact (closed and bounded), while ℝ is not compact (unbounded).

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