# Relationship between seminormed, normed, spaces and Kolmogrov top. spaces

 P: 424 For the => direction you are on the right track. You want an open ball B with center x, but with radius small enough that y is not in B. d(x,y) seems to be the only number you have to work with so try choosing a radius based on that. If you can find a radius such that y is not in B, then you are done because open balls are open sets. In the other direction you wish to show $\|x\| = 0$ implies x=0. Suppose $x \not= 0$, then by the Kolmogorov condition you can find an open neighborhood U of either x or 0 which does not contain the other. Try to use this to find a radius r such that B(x,r) or B(0,r) is contained in U, in which case you can show $\|x\|\not=0$. EDIT: Here B(p,r) means the open ball with center p and radius r.