Hi! I don't understand a part of the proof.
The theorem states:
Let X be a normal space; let A be a closed subspace of X.
(a) Any continuous map of A into the closed interval [a,b] of R may be extended to a continuous map of all of X into [a,b].
(b) Any continuous map of A into R may be extended to a continuous map of all of X into R.
My question is about part (b).
Let f be a continuous map from A into (-1,1).
It then states that "The half of the Tietze theorem already proved shows that we can extend f to a continuous map g:X --> [-1,1] mapping X into the closed interval".
How does part (a) show this?
(a) tells us that f should be a continuous map from A into the closed interval [-1,1] so that we can extend it to a continuous map g from the entire space X into [-1,1]. It doesn't say anything about A being mapped into open intervals. What's the connection I'm missing?