Equivalent distances define same topology

Question

I have to prove that

equivalent distances define same topology.

I know there are similar questions, so please don't have a go at me but I am still confused and they don't answer it in the way I have been taught.

If distances are equivalent then there exist an $\alpha$ and $\beta$ more than zero such that $$\alpha d_1(x,y) \leq d_2(x,y) \leq \beta d_1(x,y)$$

Please help me

What is the way you were taught? Do you want to use balls or sequences or something else? // You didn't use $\beta$ in the definition of equivalence of metrics. // To get $d_1$ and $d_2$ use d_{1} and d_{2}. — Martin, May 05 '13 at 01:00
I have to prove equivalent distances define the same topology — andwil, May 05 '13 at 01:01
The above is my definition for equivalent distances... i'm assuming that i have an $\epsilon \greater 0$ and a $\delta \greater 0$ — andwil, May 05 '13 at 01:02
Ok what my lecturer told me to start with was $d1 < \epsilon$ and $d2 < \delta$ then prove delta is less than epsilon and then vice versa — andwil, May 05 '13 at 01:08

score 4 · Answer 1 · answered May 05 '13 at 01:08

4

Assume that $U$ is $d_1$-open. It means that for any $x$ there is an $r>0$ such that $B_x^{d_1}(r)\subseteq U$. But we have $B_x^{d_2}(r/\beta)\subseteq B_x^{d_1}(r)$ by assumption.

The other direction goes similarly, using $\alpha$.

answered May 05 '13 at 01:08

Berci

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Equivalent distances define same topology

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