Topology: An Inquiry-Based Approach

We have seen that we can make a set into a metric space with different metrics. For example, the spaces $({\mathbb{R}}^2,d_E)\text{,}$ $({\mathbb{R}}^2,d_T)\text{,}$ $({\mathbb{R}}^2,d_M)\text{,}$ and $({\mathbb{R}}^2,d)$ are all metric spaces, where $d_E$ is the Euclidean metric, $d_T$ the taxicab metric, $d_M$ the max metric, and $d$ the discrete metric. But are these metric spaces really “different” metric spaces? What do we mean by “different”?

If there is a bijection between metric spaces that preserves distances, we say that the metric spaces are metrically equivalent.

Any function $f$ that preserves distances (like the one in Definition 14.5) is called an isometry.

Metric equivalence is a very strong type of equivalence — the existence of an isometry does not allow for much flexibility since distances must be preserved. From a topological perspective, we are only concerned about the open sets — there are no distances. The open unit ball in $({\mathbb{R}}^2,d_E)$ and the open ball in $({\mathbb{R}}^2,d_M)$ (where $d_E$ is the Euclidean metric and $d_M$ is the max metric) are not that different as we can see in Figure 14.7. If we don’t worry about preserving distances, we can stretch the open ball $B_E=B((0,0),1)$ in $({\mathbb{R}}^2,d_E)$ along the lines $y=x$ and $y=-x$ uniformly in a way to mold it onto the unit ball $B_M=B((0,0),1)$ in $({\mathbb{R}}^2,d_M)\text{.}$ The important thing is that this stretching will preserve the open sets. This is a much more forgiving type of equivalence and maintains the central idea of topology that we have discussed — what properties of a space are not altered by stretching and bending the space. This type of equivalence that allows us to manipulate a space without fundamentally changing the open sets is called topological equivalence.