Is connectedness varies with change of metric in metric topology

Question

I am confused about a fact of topology that " connectedness varies as the induced metric changes on a metric topology"

I hope answer should be no . But I want a proper justification. Please help.

Answer 1

$\mathbb{R}$ is connected in its standard topology induced by its usual metric. But $\mathbb{R}$ is disconnected in the discrete topology (any subset is open), which is induced by the discrete metric $\rho:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ given by $\rho(x,y)=0$ if $x\neq y$ and $\rho(x,x)=1$ for all $x,y\in \mathbb{R}.$

Answer 2

Connectednes is a topological property. It solely depends on the topology of the space. So, if you change your metric the topology induced by new metric may be different from the topology induced by the previous metric. So, a connected space may become disconnected in the metric and vice-versa.