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 $X$"?

(1) Let $X$ be the real line, topologized so that a subset $U$ is open if and only if either (i) $U=\emptyset$ or (ii) $0\in U$. The singleton $\{0\}$ is dense in $X$, even though $X$ is far from finite.
(2) We can also say that $x$ is an isolated point of $X$ if and only if $\{x\}$ is open in $X$. The bit about the ball could be better stated to say that "there is an open ball centered on $x$ that has only $x$ as an element"--that is, $\{x\}$ is an open ball. For an example of how this can happen in a metric space, let $X$ be any non-empty set you like, and define $d:X\times X\to\Bbb R$ by $$d(x,y)=\begin{cases}0 & x=y\\ 1 & \text{otherwise.}\end{cases}$$ This is a metric on $X$ called the discrete metric. It has the property that for any $x\in X$ and any real $0<r<1$, the ball centered at $x$ of radius $r$ is simply the singleton $\{x\}$--so all points of $X$ are isolated when considered in the topology induced by this metric.
