There is a one-to-one and onto mapping between any two intervals. The intervals can be of different sizes, infinite, open, closed, or half-open. By composing the mappings shown here, sometimes with slight modifications, you can find a mapping between any two types. Therefore any two intervals have the same number of points. In fancier language, any two intervals are equipotent and have the same cardinality as that of the real numbers.
The last case (half-open to open), is not a continuous map and has a countable infinity of jumps near the origin.