# Confused about the basis of a topology space

Say, for example, let $$(X, d)$$ be a metric space, then the basis of the topology which the metric d induced is open balls $$\mathbb{B} = \{\mathbb{B}_d(x, r), x\in X, r>0)$$.

Then those open balls cover the whole $$X$$.

Since there exists a ball $$\mathbb{B}(x, 1)$$, such that $$x\in \mathbb{B}(x, 1)$$, for every $$x\in X$$.

But how could we know, every such open ball $$\mathbb{B}(x, 1)$$ is actually in $$X$$? Can it possible that some such open balls are only partial in $$X$$? Or what open in open ball here just means open in the subspace $$X$$?

• By definition of open ball $\mathbb{B}_d(x,r):=\{y\in X\text{ such that }||y-x||_d<r\}$ it's a subset of $X$ so it is properly contained in it. – Sebastian Cor Nov 21 '19 at 3:08
• @SebastianCor But for example let [0, 1) be a metric space induced by R usual metric on [0, 1), then every ball containing 0 is not properly contained in [0, 1) – Cathy Nov 21 '19 at 3:24
• Considering $[0,1)$ as its own space, every ball containing $0$ is properly contained in $[0,1)$. For example, $\mathbb B(0,1) = \{x\in[0,1):|x|<1\} = [0,1)$. – Math1000 Nov 21 '19 at 3:44
• @Math1000 So the B(0,1) is open somehow means open in the subsapce [0, 1), not the whole space, say such as R? – Cathy Nov 21 '19 at 3:48
• Re: Last comment above. Exactly. There is no "whole space" in the sense that $any$ metric space is a proper subspace of a larger metric space. – DanielWainfleet Nov 21 '19 at 3:53

$$\mathbb{B}_d(x,r) \;\; =\;\; \{p \in X \; | \; d(x,p) < r\}$$
hence by definition we have that $$\mathbb{B}_d(x,r) \subseteq X$$. Now, if you are in some metric space $$X$$, you may be interested in working in some subspace $$Y$$ of $$X$$, in which open balls in $$Y$$ will be those induced by $$X$$. In other words, we will have
$$\mathbb{B}_d^Y(x,r) \;\; =\;\; \{p \in Y \; | \; d(p,x) < r\} \;\; =\;\; \mathbb{B}_d^X(x,r) \cap Y.$$
It's therefore important that when you talk about open balls that you are clear about whether they are open balls with respect to the total metric space $$(X,d)$$ of interest, or if you're referring to a subspace $$Y$$ of $$X$$.