Topology Definition and 798 Threads

ABAQUS provides geometric restrictions such as a planer, rotational, and other symmetric, but there is no axis symmetric restriction. I know that the 2D axis symmetric element model could be possible to make in PART section. But I want to know that a full 3D element model could be optimized by...

I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts. The Bendixson criterion is a theorem that permits one to establish...

Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...

Sketch of proof: ##1.## Let ##V## be open in ##Y##. ##2.## For arbitrary ##f:X\longrightarrow Y## and for arbitrary ##V##, ##f^{-1}(V)## is in ##X##. ##3.## ##f:X\longrightarrow Y## is continuous, so ##f^{-1}(V)## is open in ##X##. ##4.## Every subset ##f^{-1}(V)## of ##X## is open, so ##X##...

Hello! I'm a physics graduate who is interested to work in Mathematical Physics. I haven't taken any specialized maths courses in undergrad, and currently I have some time to self-learn. I have finished studying Real Analysis from "Understanding Analysis - Stephen Abbott" and I'm currently...

Hello! I have two related exercises I need help with 1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then...

Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...

I included this image because it is easier than typing it out. Anyway, this is an old problem I need to catch up on. I have a clue as to how to do part a. I could say given an x that is a member of ∩V(Ai) which implies that x is a member of V(Ai) for ∀i. Then we can say ∀i all polynomials are in...

Hello : doing some reading in physics and some of it is in solid state physics , in Ashcroft- mermin book chapter 2 page 33 you read " Thus if our metal is one dimensional we would simply replace the line from 0 to L to which the electron were confined by a circle of circumference L. In...

I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...

Hey! :giggle: We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$. We say that $U\subset X$ is open if: (a) For each point $x\in U\cap \mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$...

Hey! :giggle: Does the sequence $x_n=\frac{1}{n}$ converges as for the cofinite topology on $\mathbb{R}$ ? If it converges,where does it converge? Could you explain to me what exactly is meant by "cofinite topology on $\mathbb{R}$" ? Do we have to define first this set and then check if we...

Ok, sorry, I am being lazy here. I am tutoring intro topology and doing some refreshers. Were given the subspace topology on [0,1] generated by intervals [a,b) and I need to answer whether under this topology, [0,1] is Hausdorff, Compact or Connected. I think my solutions work , but I am looking...

Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...

I wonder if anybody has an idea for a topology on the set of Lie algebras of a given finite dimension which is not defined via the structure constants. This condition is crucial, as I want to keep as many algebraic properties as possible, e.g. solvability, center, dimension. In the best case the...

I do not understand what is to verify here. The problem already defined what it means to be a trivial and discrete topology but it did not state what it means to be "weak" and "strong". I assume the problem wants me to connect "weak" with trivial topology and "strong" with discrete topology, but...

Do you think a first course in analysis should focus entirely on inequalities and leave set-theoretic topology for another occasion? Should this depend on whether or not the student had a first rigorous calculus course first? If I'm not mistaken, Victor Bryant (Yet Another Introduction to...

Summary:: Subset of Codomain is Superset of Image of Preimage, and similar proof for subset of domain I was having a hard time doing the intro chapter's exercises in Munkres' Topology text when last I worked on it, and I just wanted to make sure that there's nothing betwixt analysis and...

[I urge the viewer to read the full post before trying to reply] I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modeled as a topological manifold (with a bunch of additional structure that's not relevant to this question). A topological manifold is...

Mostly I need to clear up a few basic things about functions and their inverses, the problem seems easy enough. Ok, so for (a) I have $$f^{-1}(f(A_0))= \left\{ f^{-1}(f(a)) | a\in A_0\right\}$$ but here I’m not certain if ##f^{-1}## is allowed to be multi-valued or not, the text says that if...

So formulating them was easy, just set ##C:=D\cup E## in (1) and set ##C:=D\cap E## in (2) to see the pattern, if ##\mathfrak{B}## is a non-empty collection of sets, the generalized laws are $$A-\bigcup_{B\in\mathfrak{B}} B = \bigcap_{B\in\mathfrak{B}}(A-B)\quad (3)$$...

Obviously the parenthetical part of the definition of ##F## means ##B\subset C## but we are not allowed to use ##\subset##. I do not know how to express implication with only union, intersection, and set minus without the side relation ##B\cap C = B\Leftrightarrow B\subset C##. This is using the...

Hi, I am a physicist interested in going through a basic analysis course. The real line, open sets, that whole thing. On my own, so I need a good selection of bibliography, ranging from those that are good references but too dense to actually read to those that are very pedagogical but tend to...

I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ... I need help in order to fully understand Example 3.2(d) ... .. Example 3.2(d) reads as follows: and...

Fred H. Croom (Principles of Topology) and Tej Bahadur Singh (Elements of Topology) define local basis (apparently) slightly differently ... Croom's definition reads as follows:... and Singh's definition reads as follows: The two definitions appear different ... ... Croom requires that each...

In AC-DC power supply modules, having non-isolated power converter topology may result in input "Line" (hot wire) been assigned to "Vss" (lower voltage rail) at output of power supply module, and N (neutral) assigned to Vdd (upper power rail). Inside single system, this usually does not results...

I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.4: Basis ... ... I need help in order to fully understand the order topology ... and specifically Example 1.4.4 ... ...Example 1.4.4 reads as follows: In order to...

I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.4: Basis ... ... I need help in order to fully understand some remarks by Singh just before he defines a sub-basis ... .. The relevant text reads as follows: To try...

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...

I only took an introductory real analysis course, and that was during the spring of last year. I apologize for the unnecessary and possibly stupid question, in any case.

I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...

Dear Everyone I am having some difficulties on exercise 2e from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets [FONT=MathJax_Math-italic]A, [FONT=MathJax_Math-italic]B, [FONT=MathJax_Math-italic]C, and...

Dear Every one, I am having some difficulties on exercise 2b and 2c from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets $A$, $B$, $C$, and $D$. If a double implication fails, determine whether one or the other one of the...

I learned in a vector calculus class that the operation of vectors is not defined. The professor mentioned it had to do with topology. How does the operation of vector subtraction relate to topology and how does topological properties prevent vector subtraction from being defined?

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows: My question is as...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...

The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...

How are topological spaces used in physics?

Hi, I'm not a really mathematician...I've a doubt about the difference between a trivial example of fiber bundle and the cartesian product space. Consider the product space ## B \times F ## : from sources I read it is an example of trivial fiber bundle with ##B## as base space and ##F## the...

Hello,I'm a freshman undergraduate physics student. I'm mainly considering theoretical physics for graduate school (condensed matter and AMO physics). There is an introductory topology course at my university which is offered by the math department. Will taking topology be useful for any...

Homework Statement Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? Homework Equations f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if...

in the tv show "The Big Bang Theory", Sheldon wrote a book called "A proof the algebraic topology can never have a non self-contradictory set of abelian groups". Is this just a random set of words that is meant to sound smart but in reality means nothing or is it accurate? If it is, what does it...

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...

Hi, I was playing this game in which you start from any cells of a 3x3 or 5x5 square and draw a line that loops through every cell in the box. The line can go only through a vertical or horizontal side (not diagonally). When you start from certain cells (problem cells), you can't reach at...

From Munkres, Topology: "A topology on a set X is a collection T of subsets of X having the following properties: (1) ∅ and X are in T . (2) The union of the elements of any subcollection of T is in T . (3) The intersection of the elements of any finite subcollection of T is in T . A set X for...

High school student here... Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...

I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe...

This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...

Have anybody seen a 3D function drawer like this one?(4:12) Since I don’t know what these 3D drawers are really called,I can’t find any.