I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying :
'every infinite and bounded part of $\mathbb{R^n}$ admit at least one accumulation point'
because for me a set is either bounded so finite or infinite so unbounded.
I don't really understand because I can accept the fact that without a metric, bounds make no sense in topology but here $\mathbb{R^n}$ is clearly known as a metric space.
thank you for your help
Answer
When we say that a set is finite or infinite, we are referring to the number of elements in the set, not to the "extent" (putting it roughly) of those elements. You can think of it in the following way. Any set, all of whose elements lie between (for example) $0$ and $1$, is bounded, because no part of the set can possibly "go to infinity". But clearly it is possible to have an infinite number of elements in such a set. For example, all reals between $0$ and $1$, or all rationals between $0$ and $1$, or simply all the numbers $\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\,\}\,$.
No comments:
Post a Comment