Two trivial questions in general topology I'd appreciate some guidance regarding the following 2 questions. Number 1 should be clear, and number 2 is more of a discussion:


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*Let $X$ be a topological space. Let $E$ be a dense subset. Can $E$ be finite without $X$ being finite? Or countable?

*Let $X$ be a topological space. I came across the following definition of "isolated point": a point $x$ which is not a limit point of $X\setminus\{x\}$. Or, for a metric space: a point $x$ such that there is an open ball centered on $x$ not containing any other point of $X$. These definitions make no sense to me: how can a ball of a topological space $X$ contain points not from $X$? It sounds like there is some kind of bigger topological space containing $X$ which is implicitly referred to. In that context, what meaning can one give to "an isolated point of $X$"?

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*Let $X = \mathbb{R}$ and define the topology $\mathcal{T} = \{\mathbb{R}, \emptyset, \{0\}\}$.  The set $\{1\}$ is dense in $(\mathbb{R}, \mathcal{T})$ (it's closure is clearly all of $\mathbb{R}$, and it's a finite set.

*Consider $X = [0,1] \cup \{2\}$ with the usual Euclidean metric.  The ball of radius $1$ about the point $2$ (in $X$) is just the set $\{2\}$.  You don't need to have a larger space in mind when you talk about the isolated points of $X$.  The ambient space $\mathbb{R}$ doesn't play any role here.
Alternatively, you could think about the metric $d(x,y) = 1$ if $x \neq y$ and $0$ otherwise on $\mathbb{R}$.  Then the ball of radius $1$ about any point in $\mathbb{R}$ is the singleton $\{x\}$.
