Topology question

• May 4th 2012, 04:30 AM
hairymclairy
Topology question
How do I show that convergence is a topological property?

Thanks.
• May 4th 2012, 05:23 AM
Plato
Re: Topology question
Quote:

Originally Posted by hairymclairy
How do I show that convergence is a topological property?

What are the exact properties of a homeomorphism?
Now think:"How are continuity and convergence related?"
• May 4th 2012, 05:26 AM
Sylvia104
Re: Topology question
Let $f:X\to Y$ be a homeomorphism between topological spaces $X$ and $Y.$ If $\left(x_n\right)_{n=1}^\infty$ is a sequence in $X$ converging to $x\in X,$ show that the sequence $\left(f(x_n)\right)_{n=1}^\infty$ in $Y$ converges to $f(x)\in Y.$

The sequence $\left(x_n\right)_{n=1}^\infty$ converges to $x\in X$ iff for every open set $U\subseteq X$ containing $x$ there exists a natural number $N$ such that $x_n\in U$ for all $n>N.$