Convergence in topological space

[Julio1]

Let $(X,\tau)$ an topological space. Show that $x_n\to_{n\to \infty} x$ if and only if $d(x_n,x)\to_{n\to \infty} 0.$

Hello, any idea for begin? Thanks.

 

[Euge]

Hi Julio,

Is $d$ supposed to be a metric? If so, the problem statement makes little sense, since there non-metrizable topological spaces.

 

[HallsofIvy]

What definition of "limit" are you using? If, as your use of "d" implies, this is a metric space, that looks pretty much like the standard definition of "limit" in a metric space.