Proposition 5

Let I be any interval in \mathbf{R}. Then the topology induced on I by the topology of \mathbf{R} is generated by the open intervals of I (where I is considered as linearly ordered by the relation x \leq y).

Every open interval of I is the trace on I of an open interval of \mathbf{R}. This is clear for a bounded interval, and the unbounded interval ]a, \to[ of I is the trace of the unbounded interval ]a, \to[ of \mathbf{R}. We may therefore restrict ourselves to the case I = \mathbf{R}; but in this case the result follows from Proposition 3, since every neighbourhood of a point x \in \mathbf{R} contains an open interval ]x - a, x + a[.