# gn.general topology – Clopen base for topological space

Leave $$(X, d)$$ a metric space, $$B (x, varepsilon): = {y in X | ~ d (x, y) < varepsilon }$$ where $$x in X$$ Y $$varepsilon> 0$$ Y $$(X, mathcal {T} _d)$$ The natural topological space for the metric space. $$(X, d)$$. Let's suppose $$B (x, varepsilon)$$ It is closed for everyone $$x in X$$ Y $$varepsilon> 0$$, so

$$1.$$ Every $$A in mathcal {T} _d$$ Is it clopen?

$$2.$$ Yes $$A$$ Is it closed then it can be written as a union of elements of the base?