I don't agree you're dumb because of the question. Why don't you tell me the definition of $\lim_{n \to \infty} x_n = a$, and then explain to me why your desired result follows from the definition?
Let's say it carefully (it is always good to be very careful/precise when learning something new, and to be less careful once you have mastered it).
You've just told me that for all $\epsilon$, there exists an $N = N(\epsilon)$ so that for all $n \geq N(\epsilon)$, we have $|x_n - a| < \epsilon$.
From there I think you want to say: "if $n \geq N(\epsilon)$, then $n + 1 > N(\epsilon)$, and so this condition holds for $n+1$ as well; we have that $|x_{n+1} - a| < \epsilon|$ for all $n \geq N(\epsilon).$
And therefore that same $N(\epsilon)$ is also a suitable $N$ for the sequence $x_{n+1}$.
I hope it's clear that there is something gained in being so precise, that it's not just pedantry. I think the point of being so precise is that you understand the why more, and it helps you generalize to other situations once you understand the reasoning.
@robjohn Good one... I've mostly trained myself to avoid ambiguity like this, after a student was deeply hurt after mishearing "And you're done".
@AlessandroCodenotti Off the top of my head, no. But it seems counterexamples to your specific situation are abundant: take the Denjoy counterexample T : S^1 -> S^1 which is topologically transitive but has an invariant Cantor set, and then put mass 1 million on the Cantor set and Lebesgue outside. This seems to be an invariant measure which is not ergodic to me
It seems to me that you can always "blowup" some generic topologically transitive map to construct examples of this sort.
$P_0$ is atmospheric pressure and P1 is height * density * gravity.
So , is the diagram correct for saying that
Total pressure in the liquid = P0 +P1. Since the direction of P0 vector in downwards?
This is somewhat surprising, because in the topological setting if $G$ acts in a topologically transitive way on a space $X$ (with some assumptions that I don't remember) and $A\subseteq X$ is $G$-invariant with the Baire property, then it is either meagre or comeagre
Ah we want $X$ to be Baire which I guess doesn't have a good translation in measure theory terms
yeh but analysts often have a different standard of rigor and I still find it hellishly complicated to formally compute anything with CW complexes. We calculated the cellular homology groups of RP^n via the local degree and when it came to actually computing the local degree we just skipped it cause its tedious calculation in local coordinates
and this "yeah it works but its tedious and obvious" happens over and over again and it was the same for the HoTT course and I really really just want some details written down carefully
@user2103480 This is a failure on the part of those teaching if they can't fill in the details
The best topologist knows how to explain every step in detail but only explains the revealing steps
Sometimes people need the details but at this point they should be getting trained to learn which details really are trivial to fill in and that requires doing them
I'm sure they can but there seems to be more of a tendency to assume that others can do that as well. And it's not like I'm generally unable to do that, but the more details are left out, the more topics are there to go through in the lecture's time so the amount of details to fill in, or learn to fill in in various areas, gets larger and larger
Of the four different lecturers in topology/geometry-related courses I attended, it was like that without exception
Interestingly, one of those lecturers also did ODE, complex & functional analysis classes that were much more stringent
Also, there's lecture notes by CW (the initials of the professor leading the research group, maybe you remember) and by another professor, alas in german
The problem is more the accumulation of things we have to check out. Our lecturer said himself before christmas that if we actually did all this in detail, we'd fill the remaining 8 weeks
Well, ok, there are a few errors and a few places where the argument doesn't hold up, but nothing beyond the standard of textbook writing in general IMO
one thing that bothered me is how he doesn't address identification maps (well, what's needed implicitly follows from the discussion of the compact-open topology in the appendix, but he doesn't mention that some of these facts are actually needed to verify that some homotopies he writes down in the early chapters are in fact continuous). perhaps it's expected the readers figures that out themselves, but he usually is more careful.
@Thorgott Identification maps, meaning quotient maps? You do need a real mastery of that to get into algebraic topology, and this is something a lot of students leaving a first course in point-set are missing, since it is treated so quickly
I make my students explain in detail how to go from his pictures that show pi_1 is a group and turn those into formulas and then a proof that the things defined in formula are continuous
Yeah, for example in chapter 0 he proves that the quotient map for a pair satisfying the homotopy extension property with the subspace being contractible, the quotient map is a homotopy equivalence. He does so by taking a homotopy $f_t$ extending a homotopy equivalence from the subspace to a point and then factors each $f_t$ individually to a quotient map $\tilde{f}_t$. These $\tilde{f}_t$ are supposed to be the final homotopy, but he doesn't justify why they are still jointly continuous in both variables (this comes down to the fact that the product of a quotient map with the identity map …
@Thorgott Aha, OK, I get your point. This is something he is indeed assuming from a point-set course (I stated that fact in mine before doing any algebraic topology, so that they could work with homotopies on T^2 in terms of homotopies on the square). I understand why he doesn't mention it even if readers aren't familiar with that fact: it points you in the wrong direction (being paranoid about the joint continuity)
@Bohemianrelativist having a unique left inverse $T' : W \to V$ means that for all $T(v) = w, T'(w) = v$, so you write $T(v_1) = T(v_2) \Rightarrow T'(T(v_1)) = T'(T(v_2)) \Rightarrow v_1 = v_2$
Means that there's some $T' : W \to V$ s.t. if $T'(w) = v$, then $T(v) = w$, so you write $ \forall w \in W, T'(w) = v \Rightarrow T(T'(w) = T(v) \Rightarrow w = T(v)$.
If a body is half submerged in liquid , the weight of the body = m1 +m2? Why do we have two masses here? Why not just M as mass of whole body.?
Total force exerted on liquid = m1g + m2g + P0 (atmospheric pressure)
Height from mass m1 and m2 is different .
@Physicsismylife you need to work on asking questions. just writing down the pieces of information that are in your mind at that moment are not sufficient. you need to think from the perspective of the person who is reading the question.
And to answer (my understanding of) your question, we divide the masses to make the computations easier: we consider the mass that is submerged and the mass that isn't. My take is the body's mass is homogeneous, so you can write the volume as a function of mass, then apply Archimedes principle to compute the forces on the object and find the state of equilibrium
But I agree with copper.hat that your questions are very poorly written, and it's hard to both make sense of them and help you because we have too little information about the context.
If a topological space $X$ can be written as a union of two open, connected, locally path-connected subsets such that the intersection of these two subsets has exactly two path-components then there is a surjective group homomorphism from the fundamental group of $X$ to $\Bbb Z$.
Last tip: I strongly advise to not include images except diagrams (and in this case, make them as neat as possible, using specialized software instead of taking pictures of what you draw if that's possible). Type out every relevant equation in Latex, don't just take a photo of what you wrote on paper
@User873110 Do you see the intuition to this problem?
I don't think local path-connectedness is a necessary hypothesis here, but maybe I'm missing something
also <insert joke about fundamental groupoids here>
no wait, obviously local path-connectedness is necessary
actually no again, it should be ok to drop local path-connectedness when one demands path-connectedness (it does break down when one only has connectedness in that scenario)
Take $a\in A$, $b\in B$, then an element of the quotient group will be of the form $(u_1u_2)^n$ where $u_1$ is a path connecting a to b in $U_1$ and $u_2$ is connecting b to a in $U_2$
Let $M$ be a Riemannian manifold and let $\nabla$ be a connection compatible with the metric. Then for any vector fields $V,W$ along the smooth curve $c:I\rightarrow M$ , we have
$\frac{d}{dt}\langle V, W\rangle =\langle \frac{DV}{dt},W\rangle+ \langle V, \frac{DW}{dt}\rangle$
However my question...
@Astyx I have a doubt that when a body is partially submerged in a liquid. Then the whole weight of body reduces because of upthrust on its surface in water .
Then why did we divide the masses in two categories m1 and m2
To fix the notation $M$ is a smooth manifold and $B$ is some bundle over it. I'm trying to understand the correspondence between connections on $B$ and sections of $J^1B$ and there are a couple of details that I'm not convinced by
Ok so to get this correspondence I want, given a connection on $B$, to build a section of $J^1B$. The way this happens is as follow: for $b\in B$ with $x=\pi(b)$ find $\sigma_b\colon U\subseteq M\to B$ a local section such that $\sigma_b(x)=b$ and such that $\mathrm{Im}T_x\sigma_b=H_b$. Then map $b\mapsto j^1_x\sigma_b$ and this is claimed to be well defined and the map I'm looking for
in solid state books, one common idea is that of constructing a reciprocal lattice. so you start with a 3D basis $\{v_1,v_2,v_3\}$ and choose $\{u_1,u_2,u_3\}$ such that $v_j \cdot u_k =C \delta_{jk}$ for some constant $C$. (typically $C=2\pi$)
However I'm failing to see both why $\sigma_b$ with the prescribed properties always exists, and why the result of this map is independent of which local section I pick
So you just need to think of $T_xH_b$ as the graph of some linear map $T_xB\to T_b F$.
So the physics literature is hiding duality stuff by abusing the cross product, I guess, Semiclassic. I'm not thinking about it too seriously right now.
$P_0$ is atmospheric pressure and P1 is height * density * gravity.
So , is the diagram correct for saying that
Total pressure in the liquid = P0 +P1. Since the direction of P0 vector in downwards?
