@gian An open cover is just a collection of open sets whose union contains any subset of the space, so the whole space itself (which is always open) is always an open cover.
The properties I know of that depend on open covers though are things like compactness which are statements about every open cover. And so the fact that these kind of trivial, finite open covers exist doesn't help much.
Could be some other applications wehre it matters though
@gian Oh thats slightly different. Hes saying that given a continuous map we can conclude that some nbhd of each point in the domain maps entirely into just one open set int he open cover. The $U_{\alpha}$ are open, so they are a nbhd for every point they contain. So there an open set containing $p$, $B_p$, contained entirely in one $U_{\alpha}$. Then $F^{-1}(B_p)$ is open in $N \times I$ so its a nbhd.
So he needs the continuity to make sure that the inverse image of an open set contained entirely in one of the $U_{\alpha}$ ahs an open preimage.
Ah I see now. But why would we want this? Why does he impose that each point in $Y \times I$ have a nbhd whose image is completely contained in some $U_\alpha$?
I am not 100% sure because what he's talking about is a bit too complicated for me to understand without reading carefully. But I think its because he only know that the $\mathscr{p}$ and their inverses are homeomorphisms when restricted to $U_{\alpha}$ so he wants to get the image of a nbhd of $(y_0, t)$ inside a $U_{\alpha}$ so then he can use the fact that the inverse of the projection is a homeomorphism to do something
but I dont know what that something is without taking way more time to read carefully @gian
sorry I dont know what the hell font that is hes using for that script 'p'
@gian Cool. This may be bad advice, I'm not an expert, but if you're having questions about some of these foundational issues about open covers and mapping into a particular open cover and such, it'd probably be worthwhile to go through a more elementary topology book before trying to tackle all of this.
This might sound strange, but could anyone recommend me a good (study) chair that doesn't cost an arm and a leg? One that you can sit on for prolonged periods without feeling uncomfortable.
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial theorems such as Krull's intersection theorem.
Recently T. Coquand and H. Lombardi have found a su...
If I have an algebraic extension $L/F$ with $\alpha$ algebraic over $F$, then degree of $L(\alpha)/L$ is less than that of $F(\alpha)/F$, right?
The minimal polynomial of $\alpha$ in $F[x]$ pushforwards to a polynomial in $L[x] \supset F[x]$ which vanishes at $\alpha$. So it's in the ideal in $L[x]$ generated by the minimal polynomial of $\alpha$ over $L$.
I mean, secretly that's what I am doing, right? The fact that minpoly of $\alpha$ over $L$ divides minpoly of $\alpha$ over $F$ implies degree of the minpoly of $\alpha$ over $L$ is less than that of over $F$
Hm. For covering spaces, if $f : X \to Y$ and $g : Y \to Z$ are covering maps then $g \circ f : X \to Z$ is one if $g$ is finite-sheeted. I wonder if a similar phenomenon happens for field extensions
Hi, I wonder if anyone has read Conlon's Differential Manifolds book?
Definitions there seem so non-constructive to me that I find it very difficult to draw any understanding of the material by reading this book, and still this is a required textbook for the class…
Isnt if the same for vector fields though? For example for an arbitrary smooth map the thing you would want to be the pushforward isnt necessarily a vector field on the codomain manifold because the image of the map might not be the whole manifold
So this is the definition of a derivative he gives. It is basically defined by the product rule property and I see no way how to deduce its normal definition from here.
@mikeonly Lee defines the derivative basically the same way. The only way to get something that looks familiar is to check what happens in local coordinates
@BalarkaSen Yes, I understand that based on what he says in the book… I think he mentions derivations somewhere later in connection to Lie derivatives.
Actually, it is even earlier than Definition 3.1.22, in Definition 2.7.3 he says that if $F$ is an associative ℝ-algebra with unity, then the linear map $\Delta \, : \, F \leftarrow F$ with the same property is a derivation of $F$.
@Liad Instead, you can also just start trying some matrices. Consider what the diagonal contributes to the characteristic polynomial (the $x^2$ and $x$ parts) and what it must look like to get the right coefficients there
@mikeonly To compute a derivative? Ya you have to go to local coordinates. Bt thats true of basically anything on manifolds. You can prove things abstractly but to do some concrete calculation you have to go to coordrinates
i got something else that im stuck on. define $f(z) = z \ ^ 2 + \lambda $ for a fixed $lambda \in \Bbb C$ . now define $V = \{x : |x| \gt 1/2 + \sqrt(5/4 + |\lambda) \}$ i have proved that $x \in V $ implies $|f(x)| \gt |x| +1$ , now i need to show that this implies that the set $F = \{x | $ there exists $R \gt 0$ s.t $|f \ ^ n(x)| \le R \}$ is bounded. i thought showing that $F \subset V \ ^ c$ but i am a bit stuck on showing that
i want to show that $x \ in V $ implies $|f \ ^ n (x) | \to \infty$
@KevinDriscoll I feel completely lost when I try to go from coordinate-free definitions Conlon gives to concrete problems in my problem sets. Very frustrating.
I could not take elementary differential geometry this term, so I am attempting graduate level differential manifolds now. Graduates feel almost the same way as me. :|
@mikeonly There I cant help you. I do physics and the way the class works for me is Im rubbish at the abstract stuff. but ask me to do some concrete computation and Im miles ahead of some of the math grads.
@mikeonly Depends on the style your course prefers. Lee is an okay balance between the abstractness and concreteness. I like Guillemin and Pollack more
@KevinDriscoll Fun stuff. I am from math physics as well, but I am taking a rest from physics this term. Our professor works in string theory, I suppose, and teaches this differential manifolds.
@KevinDriscoll Introduction to Smooth Manifolds.
And the prof really like abstract nonsense I would say. But it is nonsense just because he cannot rigorously explain it, I feel like.
@mikeonly Im not sure if this is said in Conlon or not, but my rule fo thumb is.... computations in local coordinates always work the way I think they should from ordinary calculus in euclidian space
derivatives, change of coordinates, all of it
Ah yea, I believe it. Im TAing a class where sometimes the prof admits he doesnt understand some piece of the material and hes teaching it precisely to understand it better. It can be quite frustrating.
@KevinDriscoll Haven't seen it in words in Conlon, but I have the same intuition which I cannot make coherent with what is taught in class.
I cannot even distinguish between the state where the prof knows so much he cannot put it in words and the state where he knows nothing and just speculates on a topic.
i need to conclude that the set $F = \{x : $ there exists $R $ s.t $|f \ ^ n(x) | \le R \}$ is bounded. i thought showing $F \subset V \ ^ c $ , but i got stuck on it
i want to show that $x \in V$ implies $|f \ ^ n(x) | \to \infty$
@mixedmath: Is it a good thing to tell about a room (of which I am one owner) in this chatroom so that it may attract more discussions on the topic for which the room is intended?
If it's a room discussing a topic related to maths there's nothing wrong with advertising it here (unless you spam it constantly), if it is a room about, say, ikebana, it might be received differently
user131753
4:00 PM
Then I will just post a link of the room with a brief description. If anyone thinks that it's inappropriate to do this, please flag it.
While reading coordinate geometry I learned a interesting fact.
If the pole of a straight line wrt a conic of parameter $a$ lies on a similar conic with parameter $a/n$ then the straight line is tangent to the similar conic with parameter $na$
Here conic refers to parabola,ellipse,hyperbola and...
While reading coordinate geometry I learned a interesting fact.
If the pole of a straight line wrt a conic of parameter $a$ lies on a similar conic with parameter $a/n$ then the straight line is tangent to the similar conic with parameter $na$
Here conic refers to parabola,ellipse,hyperbola and...