clerification of definition topology

Question

I have read that [0,1] with respect to the metric topology is a topological space. Where the metric topology is that a subset of X=[0,1] is open (and in the topology) if and only if it is a union of balls. In the definiton of a topology it says that X needs to be in the topology, i.e X needs to be open in X. But [0,1] can not be written as a union of balls. Therefore i am wondering wether you can take any set X and put a topology on it, and define X as open, even thoug it is not open w.r.t the "topology rule", or if i am missunderstanding something, and [0,1] actually can be written as an union if balls?

Answer 1

In the interval $[0,1]$, the open ball with center at $\frac12$ and radius $1$ is actually $[0,1]$. By definition, this open ball consists of all points $x\in X$ with $|x-\frac12|<1$, and this is clearly true for all elements of $X$.

From the fact that $X\subseteq \Bbb R$, one might intuitively expect that ball to be $(-\frac12, \frac32)$. This is not true. Open balls in a metric space are always subsets of the metric space themselves, so open balls of $X$ cannot go outside $X$.

If you want open balls in some metric space which lies naturally inside some bigger metric space, you have to be very clear on whether those balls are considered balls in the larger space or in the smaller space, because that does affect what the balls look like once they get close to the boundary of the smaller space (basically, is the smaller space thought of as a space on its own, or is it just a subset of the larger space). In this case, we're forgetting $\Bbb R$ and focusing on $X$, so that's where the open balls lie.