# Thread: metric space

1. ## metric space

Hello

2. Originally Posted by patricia-donnelly Let (X,d) be a metric space and say A is a subset of X. If x is an accumulation point of A, prove that every r-neighbourhood of x actually contains an infinite number of distinct points of A (where r>0).
Suppose $\displaystyle x$ is an accumulation point of a set $\displaystyle A$. Then there exists distinct $\displaystyle x_n \in A$ such that $\displaystyle x_n \to x$. Let $\displaystyle r>0$ then by definition of convergence it means $\displaystyle d(x,x_n) < r$ for $\displaystyle n\geq N$. Thus, $\displaystyle x_N,x_{N+1},...$ are distinct elements which lie in $\displaystyle B(x,r)$ and hence there are infinitely many.

Using this, prove that any finite subset of X is closed.
A set is closed if and only if it contains all its accumulation points. A finite set clearly has no accumulation points, hence the set is empty. And it contains the empty set.

3. You've been a great help. Thank you very much

#### Search Tags

metric, space 