Question

Prove or disprove
a. Every topological space is metrizible
b. Any metric defined on X(#0) induces a topology on x

Answer 1

1) Non-normal spaces cannot be metrizable. It is imply from statement that any metric space is normal (see [1]). Example of non-metrizable space: the topological vector space $\{f\colon\mathbb R\to\mathbb R\}$ with the topology of pointwise convergence.

Particularly,  If a space is not Hausdorff it's not a metric space. Example of non Hausdorff space is infinite $X$ with topology$\tau = \{ U \subset X\colon U = \emptyset \text{ or } X\setminus U \text{ is countable}\}$

2) For any metric space $(X,d)$ the collection of subsets $B(x,r)=\{y\in X\colon d(x,y)<r\}$ form a basis for a topology on $X$. It is obviously due to fact that an intersection of balls have another ball (see [2], first Lemma).

[1] https://math.stackexchange.com/questions/2872410/metric-space-is-normal-proof

[2] http://sites.millersville.edu/bikenaga/topology/notes/metric-spaces/metric-spaces.html