Suppose $\tau_1$ and $\tau_2$ are topologies on $X$ defined by $d$ and $p $,respectively which of the following statement is True

Question

let $(\ X,d)$ be a metric space . consider the metric $ p$ on $X$ defined by $\ p(x,y) = min \{\frac{1}{2}, d(x,y) \}, x,y \in X$. Suppose $\tau_1$ and $\tau_2$ are topologies on $X$ defined by $d$ and $p $,respectively

which of the following statement is True ?

$a)$ $\tau_{1}$ is a proper subset of $\tau_{2}$

$b)$$\tau_{2}$ is a proper subset of $\tau_{1}$

$c)$ neither $\tau_{1} \subseteq \tau_{2}$ nor $\tau_{2} \subseteq \tau_{1}$

$d)$$\tau_{1} =\tau_2$

My attempts : i thinks option c ) will correct

is it true ?

Answer 1

The correct answer is d).

Indeed, recall how you generate topology from a metric: $U$ is open if for any $x\in U$ there is $r>0$ such that an open ball $B(x,r)$ centered at $x$ of radius $r$ is fully contained in $U$. Or shortly: $B(x,r)\subseteq U$. Open ball in the given metric of course.

Now all you have to see is that for $0<r<\frac{1}{2}$ open balls in both metrics coincide.