## List of classes of functions which provide “homomorphisms”. [closed]

I recently learned, through my Topology coursework, that for any topological space $\mathbb{X}$, subset $A \subset \mathbb{X}$ and any quotient map $p : \mathbb{X} \rightarrow A$ that $$p^{-1}(\cup_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cup_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ $$p^{-1}(\cap_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cap_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ for indexed collections $\Lambda$ of open sets $\mathcal{U}$ in the quotient topology on $A$. It is true that similar facts hold for closed sets. So there is a homomorphism over set union and intersection operations. What is especially interesting to me is how broadly this homomorphism applies - it is respected for any surjective function $p$.

I am looking for similar examples of homomorphisms. These homomorphisms do not need to be over 'usual' ring operations. The best examples are homomorphisms which are true for large classes of functions.

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I'm not quite sure what you mean by a homomorphism - do you mean a morphism of Boolean algebras? If you are just after examples of "similar things happening in mathematics", i.e. maps preserving certain structure, then you can more or less throw a brick with your eyes shut and hit something fitting the description... – Yemon Choi Apr 15 2011 at 2:13
I have renamed the question because it was obscuring my meaning slightly. It is my understanding that a function "f" is a homomorphism over some operation "*" when f(a*b) = f(a)*f(b). I am looking for something a little bit more. I am looking not for specific functions but for classes of functions for which homomorphisms hold true. – Ross Snider Apr 15 2011 at 2:27
The equations you give have nothing to do with topology. They hold for any function on any set. See en.wikipedia.org/wiki/… – lhf Apr 15 2011 at 2:34
@Ihf :-) This additional information is in fact good to know. I should have mentioned some of how it ties into topology more closely. By definition the image of open sets in X of p generate the quotient topology on A and it holds (with the identities you linked to and the definition of open and closed sets) easily that the closedness and openess of the image and preimage are the same. It is topological, just not as much as it first appeared to me. – Ross Snider Apr 15 2011 at 2:41
"Homomorphism" is typically only used in an algebraic setting. Here you might say something like, "Taking inverse images commutes with arbitrary unions and intersections." – Kevin Ventullo Apr 15 2011 at 2:41

## closed as too localized by Andres Caicedo, Daniel Moskovich, Andy Putman, Pete L. Clark, Qiaochu YuanApr 15 2011 at 4:20

A similar phenomenon occurs in measure theory. Consider subsets of some fixed set $\Omega$. A collection $R$ of subsets is called a ring if

$\emptyset \in R$,

$A,B\in R \Rightarrow A\backslash B \in R$, and

$A,B\in R \Rightarrow A\cup B \in R$.

A ring is called an algebra if $\Omega \in R$ also (this gives all complements, not just relative complements). You also get intersections because $A\cap B = A\backslash (A\backslash B)$. If you've ever heard the term $\sigma$-algebra, that is also an algebra, but closed under countable union.

Now for the "homomorphisms." A premeasure is any function $\mu$ from a ring $R$ to the non-negative real numbers with $\mu(\emptyset)=0$. A premeasure is called a content if for every sequence $(A_n)$ of pairwise disjoint sets of $R$,

$$\mu(\cup_{i=1}^n A_i) = \sum_{i=1}^n{\mu(A_i)}$$

From my reading of your post, this seems to be exactly the property of being a homomorphism'' you are looking for. If I'm mistaken, please let me know and I can probably come up with a different example. Note: if $R$ is a $\sigma$-algebra, then a premeasure is called a measure, which is the start of measure theory.

A lot of people seem to struggle with measure theory the first time they take it (I know I did!), and I think one of the reasons is that $\sigma$-algebras are often unmotivated and introduced without examples (e.g. in Big Rudin). I prefer the treatment above, which I learned from a great book called Elements of Measure Theory by Heinz Bauer.

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I can't resist adding one more fact. Now that we have a notion of ring, we can also get a notion of ideal. A subset $I$ of a ring $R$ is called an ideal if $\emptyset \in I$, $N\in I, M\subset N, M\in R \Rightarrow M\in I$, and $M,N\in I \Rightarrow M\cup N\in I$ If you also have countable unions then you get a $\sigma$-ideal. If you like this way of characterizing subsets of the power set, you can also look into Dynkin Systems, e.g. at en.wikipedia.org/wiki/Dynkin_system – David White Apr 15 2011 at 2:53
David. This is a lot of information to take in. It does seem to be an example of the sort of property (which I guess isn't called a homomorphism) I am looking for. I'm actually confused about the property which defines 'a content'. It seems to me like A_i are sets of R but that you are summing them, while mu's domain is the nonnegative reals... Supposing I've just missed something (which is always the case) I doubt the example will help me with my application but it sure is interesting! Also, I'm taking Topology without having taken Reals (in class students keep talking about Rudin in Reals). – Ross Snider Apr 15 2011 at 3:01
@Ross Snider: note that MathOverflow is a Q&A site for research level mathematics, i.e., the mathematics that shows up in advanced graduate courses and beyond. Your question seems more basic than that. It's more than fine to ask basic questions, but there's a different site for that: math.stackexchange.com. I suspect that your question will soon be closed, at which point you might want to take a deep breath, think about how to reformulate it, and ask it at the other site mentioned above. – Pete L. Clark Apr 15 2011 at 3:31
Ross, I understand what you mean. This material usually takes at least a week to cover in an undergraduate course. The content is a function from $R$ to $\mathbb{R}_{\geq 0}$, i.e. it has range the non-negative reals. Probably the hardest thing to wrap your mind around is the fact that the domain is contained in the power set of $\Omega$, i.e. you have equations like $\mu(\{1,2,5\}) = 15$ and $\mu($even integers$) = 9$. As for Rudin, the text is called Real and Complex Analysis: amazon.com/… – David White Apr 15 2011 at 3:32
Ross, in case this gets closed, I thought I'd give you one more thing to look into. Your comment above about Boolean logic makes me think you could learn something from lattices: en.wikipedia.org/wiki/Lattice_%28order%29 In a collection of sets, union acts like join and intersection is meet. Your "homomorphisms" are maps $f$ such that $f^{-1}$ preserves arbitrary meets and joins. As you progress in math you'll see that $f^{-1}$ often preserves everything nice, whereas $f$ fails, e.g. $f(A\cup B)\neq f(A)\cup f(B)$ could occur. Anyway, AND and OR can also been seen as a join and a meet. – David White Apr 15 2011 at 3:42