Question

Prove that [0,1] and (0,1) are not homeomorphic by showing that any continuous map f:[0,1] → (0,1) is not onto.

Similarly, prove that [0,1),[0,1],(0,1) are not homeomorphic to each other.

Answer 1

To prove that [0,1] and (0,1) are not homeomorphic, we can show that any continuous map from [0,1] to (0,1) is not onto. Similarly, [0,1), [0,1], and (0,1) are not homeomorphic to each other.

Step-by-step explanation:

To prove that [0,1] and (0,1) are not homeomorphic, we can show that any continuous map from [0,1] to (0,1) is not onto. Intuitively, this means that there is no way to map all points in [0,1] onto all points in (0,1) using a continuous function.

Suppose f:[0,1] → (0,1) is continuous and onto. Since (0,1) is open, the preimage of any open set in (0,1) must also be open. However, the preimage of (0,1) under f is [0,1], which is closed. This contradicts the requirement that f must be continuous.

Similarly, we can use similar reasoning to show that [0,1), [0,1], and (0,1) are not homeomorphic to each other. By considering the boundary points of these sets, we can show that their topological properties are different, preventing a homeomorphism between them.