# Mappings on partitive sets

1. Jul 23, 2010

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 $\sum_{i = 1}^3 \frac{3!}{i!}$.

ii) How many open mappings are there?

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: Jul 23, 2010
2. Jul 23, 2010