Thread: To show metrizable

Originally Posted by math.dj

Show that product topology on Rd*R is metrizable..

(Rd is discrete topology on R.)

A finite products of metrizable spaces is metrizable. A discrete topology on R is metrizable since it is induced by a discrete metric on R. A standard topology on R is also metrizable since it is induced by a standard metric on R. You can also use some metrization theorems here. You can use a Nagata-Smirnov metrization theorem to show that a discrete topology on R is metrizable and use a Urysohn metrization theorem to show that a standard topology on R is metrizable.

We have yet not come across Nagata-Smirnov metrization theorem and Urysohn metrization theorem ..so i can't use them..can u please tell me how do i show that R with usual topology is metrizable..i simply don't know how to write it..


thank you for your help

Originally Posted by math.dj

We have yet not come across Nagata-Smirnov metrization theorem and Urysohn metrization theorem ..so i can't use them..can u please tell me how do i show that R with usual topology is metrizable..i simply don't know how to write it..


thank you for your help

OK. It is straightfoward to see that a discrete metric on R induces a discrete topology on R. So you don't actually need Nagata-Smirnov metrization theorem.

Anyhow, to show that usual topology on R is metrizable, I recommend you to use Urysohn metrization theorem. It says that "Every second-countable regular Hausdorff space is metrizable." You need to verify that a standard(usual) topology on R is second countable and regular Hausdorff.