6.4. Topology and Continuity
Is it true that if f
is continuous, then the image of an open
set is again open ? How about the image of a closed set ?
This is true for inverse images but not for images. Consider the
example of a parabola, which certainly represents a continuous
f(x) = x2
Then the image of the set (-1, 1)
is the set [0, 1)
. That set
is neither open nor closed; in particular, it is not open.
To find a counterexample for images of closed sets, let's look
at the following function:
This function is continuous on the whole real line, and the image
of the set [0, )
is the set (0, 1]
. Therefore we have found a closed set whose
image under a continuous function is not closed (nor open).