Analysis: Continuous open mappings.

[futurebird]

Here is a mystifying question from Rudin Chapter 4, #15

Call a mapping of X into Y "open" if f(V) is an open set in Y whenever V is an open set in X. Prove that every continuous open mapping of Reals into the Reals is monotonic.​

I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... $$f(x)= x^{2}$$ map open sets to open sets? And $$f(x)= x^{2}$$ isn't monotonic on the Reals. Can someone tell me why $$f(x)= x^{2}$$ isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.

 

[morphism]

I'm having trouble proving this, in part, because I don't even think it's true. Wouldn't say... $$f(x)= x^{2}$$ map open sets to open sets? And $$f(x)= x^{2}$$ isn't monotonic on the Reals. Can someone tell me why $$f(x)= x^{2}$$ isn't a continuous open mapping of Reals into the Reals that is NOT monotonic.

What is f((-1,1))?

 

[futurebird]

What is f((-1,1))?

[0, 1)

THANKS. Got it now.