From what I have read online and from what I have read in Rudin, a collection of open sets $\lbrace$$V_{n}$$\rbrace$ is said to be a base for a metric space $X$ if every open set in $X$ can be expressed as a union of a subcollection of $\lbrace$$V_{n}$$\rbrace$. In other words, I am taking it to mean that if you give me an open set in X, I can express it as the union of some of those sets.
What will be some examples of bases for some Metric spaces?
For instance, what's a basis for the set $0\cup$$\lbrace$$\frac{1}{n}$ : $n \in$ $\mathbb{N}$$\rbrace$?
Can I write All neighborhoods with radius bigger than 0 as a basis?
How about a basis for $\mathbb{R}$? Can I write all positive intervals or the same basis I just named above? I just need a few concrete examples of some bases to get the understanding I think.
The basis of $\mathbb{R}$ with standard topology is all open intervals. If $\{B_\alpha\}$ by the basis for $X$ and $Y\subset X$ is a subspace, then $\{B_\alpha\cap Y\}$ is the basis for $Y$.
And for metric space, all open ball with radius bigger than 0 is a basis.