Prove that the following spaces are complete/not How to show if the following facts are true/false:


*

*$(0,1)$ with the usual topology admits a metric which is complete.

*$ [0,1]$ with the usual topology admits a metric which is not complete.


Since usual metric  gives usual topology and  we know  that $(0,1)$ is not complete with usual metric so the first question is false.
Since usual metric  gives usual topology and  we know  that $[0,1]$ is  complete with usual metric so the second  question is also false.
Are the justifications correct?Please help

The important thing to remember is that it is possible that a metrizable topological space $(X,\tau)$ supports two different metrics $d_1,d_2$ that induce the topology $\tau$ for which $(X,d_1)$ is complete but $(X,d_2)$ isn't. 
For the first question, note that $(0,1)$ is homeomorphic to $\mathbb{R}$. Choose some homeomorphism $f \colon (0,1) \rightarrow \mathbb{R}$ and define a metric $d$ on $(0,1)$ by $d(x,y) = |f(x) - f(y)|$. Since $f$ is a homeomorphism, $d$ induces the usual topology on $(0,1)$ but $f$ is an isometry and so $d$ is complete.
For the second question, note that $[0,1]$ with the usual topology is compact and any compact metric space is complete.
