Question

Prove or disprove
1) every topological space is metrizable
2) any metric defined on X(is not equal to 0) induces a topology on X

Answer 1

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

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 intersection of balls has another ball ([2]).

References:

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

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