Is my solution of the following problem correct?

Let $ S $ the two-point topological space in which only one of the singleton subsets is open. Prove that continuous maps from a space $ B $ at $ S $ correspond bijectively to open subsets of $ B $.

Let $ S = { star, bullet } $ and let $ { bullet } $ to be the singleton subset that is open.

Set the map $ F $ of maps $ B to S $ open subsets of $ B $ by sending a card cts $ phi: B to S $ to the subset $ phi ^ {- 1} ( { bullet }) $. This is an open subset of $ B $ because $ phi $ is continuous.

Define the map in the opposite direction by assigning to an open subset $ U $ of $ B $ function $ psi_U: B to S $ who sends each element of $ U $ at $ bullet $ and each element outside of $ U $ at $ star $. This map is continuous because the pre-image of each open subset is open: the pre-images of $ emptyset $ and $ S $ are, respectively $ emptyset $ and $ B $ (which are open), and the preimage of $ { bullet } $ is the open subset $ U $.

Let us check that $ FG = $ 1 and $ GF = $ 1.

1) $ F (G (U)) = F ( psi_U) = psi_U ^ {- 1} ( { bullet }) = U $.

2) $ G (F ( phi: B to S)) = G ( phi ^ {- 1} ( { bullet })) = psi _ { phi ^ {- 1} ( { bullet })} $. To see that $ phi = psi _ { phi ^ {- 1} ( { bullet })} $, Note that $ psi _ { phi ^ {- 1} ( { bullet })} $ send to $ bullet $ these and only these elements $ x $ For who $ x in phi ^ {- 1} ( { bullet }) $ (that is to say, for which $ phi (x) in { bullet } $, that is, for which $ phi (x) = bullet $) – That's it, $ psi _ { phi ^ {- 1} ( { bullet })} $ send to $ bullet $ precisely the same elements that $ phi $ send to $ bullet $, and $ psi _ { phi ^ {- 1} ( { bullet })} $ sends the other elements to $ star $ (which is valid for $ phi $ as well as). So $ phi = psi _ { phi ^ {- 1} ( { bullet })} $.