Topology: An Inquiry-Based Approach

If we think of a homeomorphism as allowing us to stretch or bend a space, it is reasonable to think that we could stretch an open interval of the form $(a,b)$ infinitely in both directions without altering the nature of the open sets. That is, we should expect that $\mathbb{R}$ with the standard topology is homeomorphic to $(a,b)$ with the subspace topology.

It is left to Exercise 4 to show that $\mathbb{R}$ is also homeomorphic to any interval of the form $(a,\mathrm{\infty })$ or $(-\mathrm{\infty },b)\text{.}$ Later we will determine if $\mathbb{R}$ is homeomorphic to intervals of the form $[a,b)\text{,}$ $(a,b]\text{,}$ $[a,\mathrm{\infty })$ or $(-\mathrm{\infty },b]\text{.}$