- #1
- 3,149
- 8
Another problem whose answer I'd like to check. Thanks in advance.
Let X = {1, 2, 3, 4} and Y = {1, 2, 3}. Let P(X) and P(Y) be the power sets of X and Y, respectively.
i) How many continuous mappings are there from the discrete topological spaces (X, P(X)) to (Y, P(Y))?
Well, I figured that every mapping we can define between these topologies is open, since for any open set in Y (i.e. any power set), the preimage must again be a power set in X, so the total number would be [itex]\sum_{i = 1}^3 \frac{3!}{i!}[/itex].
ii) How many open mappings are there?
The same answer.
iii) Is the identity mapping f(x) = x continuous, and if so, is it a homeomorphism?
Here I'm a bit confused, since we can't map 4 to 4, since 4 is not an element of Y. Shouldn't all the elements of the domain X be mapped into some element of Y?
Let X = {1, 2, 3, 4} and Y = {1, 2, 3}. Let P(X) and P(Y) be the power sets of X and Y, respectively.
i) How many continuous mappings are there from the discrete topological spaces (X, P(X)) to (Y, P(Y))?
Well, I figured that every mapping we can define between these topologies is open, since for any open set in Y (i.e. any power set), the preimage must again be a power set in X, so the total number would be [itex]\sum_{i = 1}^3 \frac{3!}{i!}[/itex].
ii) How many open mappings are there?
The same answer.
iii) Is the identity mapping f(x) = x continuous, and if so, is it a homeomorphism?
Here I'm a bit confused, since we can't map 4 to 4, since 4 is not an element of Y. Shouldn't all the elements of the domain X be mapped into some element of Y?
Last edited: