Homeomorphisms Between Intervals

Topology · Axiom Academy

EXAMPLE Interval Homeomorphisms Worked examples showing intervals are homeomorphic Problem 1: Construct (0,1) ≅ ℝ This function takes values in (0,1) and maps them to all real numbers. Surjectivity: For any , we need to show there exists such that . We need , which gives . Adding , we get . Thus f is surjective. Injectivity: The tangent function is strictly increasing on , so f is strictly increasing on (0,1). Therefore, if , then . Thus f is injective. Continuity of f: The function is continuous on , which includes the domain for . Composing continuous functions gives a continuous function, so f is continuous. Continuity of f⁻¹: The inverse function is: Since arctan is continuous on ℝ and division and addition by constants preserve continuity, is continuous. We use a simple linear scaling and translation. Define by: This function linearly maps the interval (0,1) to (a,b): Injectivity: Since , the function g is strictly increasing. If , then . Surjectivity: For any , we can solve for x: Since , we have and , which gives . Continuity: The function is a polynomial (in fact, linear), so it's continuous everywhere, including on (0,1). Continuity of g⁻¹: The inverse function is: This is also a linear function, hence continuous on (a,b). Problem 3: Prove [0,1] is NOT homeomorphic to (0,1)

This is the written version of the interactive lesson above. See the full Topology course.