Can someone please describe the standard topology on the closed interval [0,1]? It seems (to me) like it should consist of the open balls, {0}, and {1} (and the union/finite intersection of these, etc.)... Thanks.
I assume you mean a subset topology.
Since $\displaystyle [0,1]$ is a subset of $\displaystyle \mathbb{R}$ it means $\displaystyle U\subseteq [0,1]$ is open iff $\displaystyle U = [0,1] \cap V$ for some open subset $\displaystyle V$ of $\displaystyle \mathbb{R}$.
Therefore, the open sets of $\displaystyle [0,1]$ under the subspace topology are $\displaystyle (a,b)$, $\displaystyle [0,a)$, $\displaystyle (a,1]$ where $\displaystyle 0<a<b<1$.
Hello,
You don't talk about balls in this dimension oO
The standard topology of [0,1] may be the subspace topology on $\displaystyle [0,1] \subset \mathbb{R}$
The standard topology on [0,1] may then be $\displaystyle \tau=\{\mathcal{O} \cap [0,1] ~ : ~ \mathcal{O} \in \tau_{\mathbb{R}} \}$
And $\displaystyle \tau_{\mathbb{R}}=\{]a,b[ ~ : ~ a<b, ~a \in \mathbb{R} \cup \{-\infty\} \text{ and } b \in \mathbb{R} \cup \{+\infty\}\}$
So $\displaystyle \tau=\{[0,a[ ~,~ ]b,1] ~,~ ]a,b[ ~:~ 0<a \le 1 \text{ and } 0 \le b < 1 \}$