Defination about metric and topological space.

Question

What is the main difference of definition of continuity between metric space and topological space?

"epsilon - delta " defination will not use in topological space but we use this in metric space. Why??

Please help me to understand. Thank you!

Answer 1

If a set $X$ is equipped with a metric $d_X$, then this metric induces a topology $\tau_X$ on $X$ and equipped with this topology $X$ is a topological space.

If similarly set $Y$ is equipped with a metric $d_Y$, then this metric induces a topology $\tau_Y$ on $Y$ and equipped with this topology $X$ is a topological space.

Then a function $f:X\to Y$ is continuous if preimages of sets in $\tau_Y$ under $f$ all are elements of $\tau_X$.

It can be proved that the epsilon-delta condition - which is completely formulated on base of the metrics - is necessary and sufficient for $f$ to be continuous.

So working in metric spaces it is already possible to speak of continuity if topologies are not introduced yet.

Quite often metric spaces are studied already before topology has been on the menu. The epsilon-delta condition is then put forward as some sort of definition making it possible to handle continuity, but actually (as already said) it is a condition necessary and sufficient.