Metrics whose sum induces a topology different from components

Question

Does there exist a metric $d$ defined as the sum of two metrics $d_1$ and $d_2$ (i.e. $d(x,y) = d_1(x,y) + d_2(x,y)$), but $d$ generates a different topology than both $d_1$ and $d_2$? If so, what are some examples of this?

Answer 1

Sure. A sequence converges with respect to the sum metric $d(x, y) = d_1(x, y) + d_2(x, y)$ iff it converges with respect to both $d_1$ and $d_2$, so it suffices to construct $d_1$ and $d_2$ such that some sequence $x_n$ converges with respect to $d_1$ but not $d_2$ and some sequence $y_n$ converges with respect to $d_2$ but not $d_1$; then neither sequence converges with respect to $d$, so the topology generated by $d$ must be different from both.

This can be done as follows. We'll take our underlying set to be $\mathbb{N}$, and we'll pick metrics on $\mathbb{N}$ by embedding in $\mathbb{N}$ into $\mathbb{R}$ in various ways and considering the induced metric. First, consider the embedding

$$f(n) = \begin{cases} 0 & \text{if } n = 1 \\ \frac{1}{n} & \text{if } 2 \mid n \\ n & \text{otherwise}\end{cases}.$$

With respect to the induced metric, the sequence $x_n = 2n$ converges to $1$, but the sequence $y_n = 2n+1$ is not Cauchy. On the other hand, consider the embedding

$$g(n) = \begin{cases} 0 & \text{if } n = 1 \\ n & \text{if } 2 \mid n \\ \frac{1}{n} & \text{otherwise}\end{cases}.$$

With respect to the induced metric, this time the sequence $y_n = 2n+1$ converges to $1$, but the sequence $x_n = 2n$ is not Cauchy. So with respect to the sum metric, neither sequence converges.