Hello;
I have a question about continuous functions;
Let A={2,4,6} and B={1,3,5}. A and B are two discrete subsets of the real line so that the topology of them is the induced topology of the real line. Define a function f from A to B such that f(2)=1, f(4)=3 and f(6)=5. Is the function f continuous.
In order to prove that a function is continuous, we need to check that the pre-image of any open set in B is open in A. I confused because there is no open set in B here, so what we can say, is it true that the function f is vacuously continuous.
Thank you in advance
Not exactly. Every subset of A or B is open because every subset of A or B is the intersection of an open set of real numbers and A or B. So for example {2,4} is the intersection of A and the open interval (1.5,4.5) (or the open set ).
- Hollywood