@Thorgott: thanks for suggesting the reference. Ted’s book also has a chapter on compactness and defines sup ||Tx||, which I think I must know.
Given a continuous map $f: \mathbb R\to \mathbb R$, suppose that B is a borel $\sigma$ algebra, how do I show that $f^{-1}(B)$ is also a borel $\sigma$ algebra on R?
I showed that $f^{-1}(B) \subset B$.
But I don’t know how to show the reverse containment. :(
If I could show that $f^{-1}(B)$ contains all open subsets of R, then I’ll be done.
is 'borel sigma algebra' here = a sigma algebra generated by some topology?
this might just be the fact that the inverse image of a topology is a topology. the algebras can be different e.g. if you give both domain and codomain the usual topology on R and let B denote the usual borel sigma algebra, but let f be a constant map, f^{-1}(B) is not B but the trivial sigma-algebra
and in particular it does not include all open subsets of R in the sense of the usual topology on R
A compact invariant set $\Gamma$ of diffeomorphism $f$ is called a basic set if it is topologically transitive and locally maximal ($\Gamma = \cap_{n \in \Bbb{Z}} f^{-n}(U)$) for some neighborhood $U$ of $\Gamma$
I am trying to understnad the locally maximal part
For anyone with a reasonably strong math background who is interested: I am trying to organize and advanced (intro graduate-level) set theory study group. We will probably use Kunen and follow the syllabus from here: people.math.wisc.edu/~miller/old/m771-11/index.html
in this article of Quanta Magazine, it says in isometric embedding of $C^k$ of flow, if k>1/3, it's a energy-conserving smooth flow, but if k<1/3, it becomes energy-dissipating turbulence. It also says in the isometric embedding of $C^r$ of any manifold with a fixed boundary, if r>3/2, the manifold can be crumpled without buckling, but if r<3/2, the manifold can only be crumpled with buckling.
but I only know how to differentiate a function a positive integer number of times k to define $C^k$-diffeomorphism. In the above article, it says k or r can be a fraction number. I wonder how to define the kth derivative when k is a fraction number.
leslie, thanks to you too. As it so happened, one of my classmates was also giving me that example (this was after my comments to you). Then, I came here and saw 1potato2potato saying the same thing. I understood. :)
@leslietownes the term started already :).
I'm the only one in class who did undergrad in engineering. Well, there's one more.
But he was in academia after UG/PG, I was not. I worked in industry related to the engineering. 😬
I found an error in the book I'm reading. They wrote that if we equip weak/Mackey topology on E, with a closed subspace G, then G gas weak/Mackey topology
But in reality they meant that E/G has weak/Mackey topology
I have a question: does the subspace G always have Mackey topology?
This is true for weak topology
@leslietownes do you know perhaps? If not, I'll just post a question. I don't suppose anyone else would.
I find these things easier to think about in the language of filters. Using the usual correspondence between nets and filters, (2) is equivalent to saying that every filter on $X$ containing $A$ has an accumulation point in $X$.
So, suppose $\overline{A}$ is not compact, and we will find a filte...
I suppose it's false in general so lets assume we are in a topological vector space
@AlessandroCodenotti hi. Do you happen to know/have a reference for this?
Are the following definitions of relative countable compactness equivalent?
$\overline{A}$ is countably compact
Every sequence in $A$ has a convergent subnet
Note that for regular spaces, the analoguous definitions are equivalent for relative compactness. Also, the equivalence always holds if...
Are the following definitions of relative countable compactness equivalent?
$\overline{A}$ is countably compact
Every sequence in $A$ has a convergent subnet
Note that for regular spaces, the analoguous definitions are equivalent for relative compactness. Also, the equivalence always holds if...
