>The definition of kernal and cosets only apply to homomorphisms and not just any map between groups, by the way. So you just need a homomorphism that has infinite kernel
Oh. This clears things up a lot.
>They are always the same size (cardinality) automatically.
Maybe I'm misunderstanding things then. There are functions in which the pre-image of some points contain more points than the pre-image of other points. An example - I know it's trivial but a function f defined on R, so that f(x) =...
Oops - looks like I wrote my comment the moment you did. Thank you for pointing out the cardinality term, that is what I should be using. My "ideal" function would be one in which the carnality of the pre-image of all points (cosets of the kernel it seems) is inifinite. Mind you, it doesn't...
By size I was talking about how many points are in the pre-image, not size in any literal way. (Just out of curiosity - what is a better term than "size" to use when talking about how many points are in a set?)
>pre-image of another point is a coset of the kernel
Oh I didn't realize this - I...
I am kind of curious what topics to read to understand this concept more.
Suppose I want to find a function f: A \rightarrow B, where if you look at the pre-image of any point b \in B, the size of the pre-image of b will be quite large. Essentially, I want to find functions that map from very...
A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default?
When trying to think of a space with no cauchy...
I think this is where I'm confused. Based on this definition: https://proofwiki.org/wiki/Definition:Measurable_Function it looks like a function is "sigma measurable" based on some sigma, where a sigma is a sigma-algebra on a set. Where does the measure come into play? I didn't get the...
This might not be the right subforum, but I was told that measure theory is very important in probability theory, so I thought maybe it belonged here.I am confused about the difference between a measure (which is a function onto \mathbb{R} that satisfies the axioms listed here...
Oh yeah, I see now I guess it wouldn't make sense for it to be mapped to anything other than R, since it's trying to capture the idea of length. Thanks for the response!
Can I ask a sort of unrelated topic - can you suggest any more of these abstractions that I could go and study about? For...
Thanks for the responses everyone! This clears up my doubts. I must say I find this version more confusing and I'm not certain I see the benefit of writing it this way over the version with 4 axioms, which seems to be much clearer (especially for someone new).
Just wanted to say thanks for the responses. Just out of curiosity - is vector space definition ever generalized so that the set its mapped to is any general ring? Why is it limited to R?
In all the topology textbooks I used in school, the open set axoims specified 4 conditions on a set S:
(i) S is open
(ii) empty set is open
(iii) arbitrary union of open sets is open
(iv) finite intersection of open sets is openI noticed on proofwiki, that (ii) is omitted. I was curious if...
Sorry, I wasn't sure of the best way to phrase this. This is a common problem I keep having.
Here's the definition of a norm:
Let E be a vector space V defined over a field F. A norm on V is a function p: V \rightarrow \mathbb{R} such that:
\forall a \in F and \forall u,b \in V:
(i) p(av)...