0

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!

Golam biswas
  • 73

1 Answers1

3

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.

drhab
  • 151,093
  • 4
    The $\varepsilon$-$\delta$ definition is historically the first "correct" definition of continuity (the very first being in terms of infinitesimals by Newton and Leibniz, if it was at all discussed as "all functions are continuous" in the early days of analysis), and later open sets were defined and this definition was shown to be equivalent to the inverse image of open definition. In the twentieth century all the generalisation were made to more general notions of "spaces". – Henno Brandsma Jul 08 '18 at 21:24
  • Thanks! But can we give defination of continuity in topology by epsilon - delta? – Golam biswas Jul 09 '18 at 07:28
  • 1
    Only if the topological spaces that serve as domain and codomain are both metrizable you could give such "definition" (I would not call it a definition in that situation). Actually if you work with topological spaces then there is no need for that as definition, because you already have a definition, which mostly is much nicer to work with. If you can drop the sometimes awkward searching for the correct $\delta$ then drop it, I would say. – drhab Jul 09 '18 at 09:10
  • Special thanks goes to @drhab – Golam biswas Jul 09 '18 at 14:45
  • You are very welcome. – drhab Jul 09 '18 at 16:44