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!
