Question

Let (X,τ) and (Y,τ') be topological spaces, Prove f:X → Y is continuous if and only if for every x ∈ X and every open set O of Y containing f(x), there exists an open set U of X containing x such that f(U) ⊆ O. @ganeshie8 @myininaya @UnkleRhaukus @RadEn @robtobey @chmvijay @Luigi0210 @eliassaab @genius12 @Vincent-Lyon.Fr @Preetha @Kainui @ehuman @wolfe8 @lucaz @INeedHelpPlease? @khadeeja @thadyoung @A_clan @tester97 @Andras @Confusionist @linh412986 @BlackLabel @leozap1 @JMark @Microrobot @zacharyf @Rubio101 @NaomiBell1997 @MeganEdward @theballer225 @link,zoro @Goldenmoon @CrayolaCrayon_

Answer 1

How's that for you @ganeshie8

Answer 2

Let O be an open set in Y we muast shwo that
\[
f^{-1}(O)
\]
is open in X.

Answer 3

please stop tagging me like wtf this is the 5th time u tagged me and its annoying af and im sure that others dont wanna be mass tagged

Answer 4

bro if ur so annoyed then dont respond to the tagging...

Answer 5

Let
\[
x\in f^{-1} (O)
\]
then
\[ f(x) \in O
\]
there is an open
\[
U \subset X, \quad | \quad f(U) \subset O
\]
Hence
\[
U \subset f^{-1} (O)
\]

Answer 6

We are done

Answer 7

Quod erat demonstrandum!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 8

Btw your answer lies here: http://en.wikipedia.org/wiki/Continuous_function_%28topology%29#Continuous_functions_between_topological_spaces It's a basic proof, as @eliassaab has demonstrated.

Answer 9

Q.E.D. looks better ;P

Answer 10

I feel like I'm on the mountain tops calling to the valleys below when I shout it out loud hahahahahaha

Answer 11

anyway, where are you getting these questions from? @FutureMathProfessor

Answer 12

Want some more?

Answer 13

ya sure? this one seemed trivial. may be one that requires some work? lol @FutureMathProfessor