I’m currently working on a problem involving continuous maps and subsets of the interval (0,1) , and I would appreciate some help.

Let α:(0,1)→R2 is a continious map that maps the subset X⊂(0,1) to α(X) . Can someone help explain why each of these statements is true or false?

a) If X is a closed subset of (0,1) then its image α(X) is a closed subset of R2 .

b) If X is an open subset of (0,1) then its image α(X) is a closed subset of R2 .

c) If X is a bounded subset of (0,1) then its image α(X) is a bounded subset of R2 .

d) If X is a compact subset of (0,1) then its image α(X) is a compact subset of R2 .

e) If Y is a closed subset of R2 then its preimage α−1(Y) is a closed subset of (0,1)

f) If Y is an open subset of R2 then its preimage α−1(Y) is an open subset of (0,1)

g) If Y is a compact subset of R2 then its preimage α−1(Y) is a compact subset of (0,1)

I have reviewed the properties of continuous maps and subsets, but I’m unsure about how they apply to these specific cases, especially in terms of closed, open, bounded, and compact sets. I would greatly appreciate detailed explanations and any references to theorems or properties that might be relevant to answering these questions.

Thank you in advance for your help!