# Thread: cardinality of continuous functions f:Z->Z

1. ## cardinality of continuous functions f:Z->Z

What is the cardinality of the set of all functions from Z to Z

Dont really know how to start.
Any hints would be appreciated

2. Originally Posted by firebio
What is the cardinality of the set of all functions from Z to Z

Dont really know how to start.
Any hints would be appreciated

Well, assuming that $\displaystyle \mathbb{Z}$ has the normal metric (topology) then it is discrete and so every function is continuous and so the answer is $\displaystyle \aleph_0^{\aleph_0}=2^{\aleph_0\aleph_0}=\mathfrak {c}$
Well, assuming that $\displaystyle \mathbb{Z}$ has the normal metric (topology) then it is discrete and so every function is continuous and so the answer is $\displaystyle \aleph_0^{\aleph_0}=2^{\aleph_0\aleph_0}=\mathfrak {c}$
It isn't about topology. It's about the fact that if you aren't doing topology then I assume your idea of continuity is that of the usual idea with $\displaystyle \delta-\varepsilon$ proofs. But, with this kind of continuity (which is what I assumed before) every function $\displaystyle f:\mathbb{Z}\to\mathbb{Z}$ is continuous and so my previous answer stands.