3

If a sequence $(x_n)$ converges to $x$ in a metric space $(X, \rho)$, and converges to $y$ in a metric space $(X, \sigma)$, does it follow that $x=y$?

If not, what would be a counterexample, preferably on the space of continuous functions $C[0,1]$.

Andreas Orthey
  • 313
  • Are $\rho$ and $\sigma$ related in some way (e.g. correspond to the same topology)? If not, there is no reason for the limits to be the same. – Robert Israel Nov 22 '17 at 02:46

1 Answers1

5

No, this is incorrect for somewhat uninteresting reasons.

Let $X = \mathbb{R}$, and let $d(x,y) = |x-y|$. Let $x_{n} \in [0,1]$ be a sequence converging to $0$ in the $d$-metric. Now let's define a new metric, $\rho$, that is almost identical $d$, except that it thinks $0$ is actually $7$ and vice versa. Let's make this precise. Let $f: \mathbb{R} \to \mathbb{R}$ be a function that swaps $0$ and $7$ and leaves all other numbers unchanged. Define $\rho(x,y) = d(f(x),f(y))$. Observe that $x_{n} \to 7$ in this metric.

The trouble here is that different metrics don't care about the way we like to draw our sets. We think $7$ is where it is because we draw the number line the Euclidean way. But it's just as legal to put the $7$ symbol at the origin and the $0$ symbol where the $7$ used to go.

Elchanan Solomon
  • 30,807
  • Nice answer. Another uninteresting example would be to take the usual metric $d_1$ on the closed disk, and let $d_2$ be the same metric but where we cut out the boundary of the disk and re-glued it back with a half-turn twist. In other words, using complex numbers notation, $d_2(r_1 e^{i\theta_1}, r_2 e^{i \theta_2})$ is computed like $d_1$ if $r_1,r_2 < 1$, but if one of the $r_i$ is equal to $1$, we add $\pi$ to $\theta_i$ to compute the distance with $d_2$. If a sequence inside the open disk converges to the boundary with $d_1$, it converges to the other side with $d_2$. – Patrick Da Silva Nov 22 '17 at 03:25
  • It would be nice to have an interesting example, like something coming from functional analysis or whatever. – Patrick Da Silva Nov 22 '17 at 03:26