How do I show that convergence is a topological property?
Thanks.
Let $\displaystyle f:X\to Y$ be a homeomorphism between topological spaces $\displaystyle X$ and $\displaystyle Y.$ If $\displaystyle \left(x_n\right)_{n=1}^\infty$ is a sequence in $\displaystyle X$ converging to $\displaystyle x\in X,$ show that the sequence $\displaystyle \left(f(x_n)\right)_{n=1}^\infty$ in $\displaystyle Y$ converges to $\displaystyle f(x)\in Y.$
The sequence $\displaystyle \left(x_n\right)_{n=1}^\infty$ converges to $\displaystyle x\in X$ iff for every open set $\displaystyle U\subseteq X$ containing $\displaystyle x$ there exists a natural number $\displaystyle N$ such that $\displaystyle x_n\in U$ for all $\displaystyle n>N.$