If you will give a page number in Taylor, I will look up what you are allowed to use.

Here is the basic idea. A component is a

**maximally connect subset**. Any component of an open set is open. Two distinct components must be disjoint otherwise they would not be maximum.

Example:

is an open set with three components.

Given any open, set each of its points must belong to one of its components. Remember they are pair-wise disjoint. Because the set of rationals is countable that collection of components must be countable.