Topology Definition and 798 Threads

Think for example of the torus as a square with the proper edges identified. Viewed like this (i.e. using the flat metric), it clearly has zero curvature everywhere. More specifically, it seems Euclid's axioms are satisfied. But however we have non-trivial topology. So what's up? Or is...

Homework Statement . Prove that a closed subset in a metric space ##(X,d)## is the boundary of an open subset if and only if it has empty interior. The attempt at a solution. I got stuck in both implications: ##\implies## Suppose ##F## is a closed subspace with ##F=\partial S## for some...

Hello :) I am looking for some books for an intro to topology and what other books I need to supplement my readings not quite sure the prereqs for topology but I am willing to learn the stuff needed thank you! P.S Physical textbooks are what I am looking for but if that's not available then...

Homework Statement We are given ##H## = {##O | \forall x, \exists a,b \in R## s.t ##x \in [a,b] \subseteqq O##}##\bigcup {\oslash}## and are asked to show that it is a topology on R Homework Equations Definition of a topological space The Attempt at a Solution I am trying to...

One of the definitions of a subbasis ##\mathcal{S}## of a set ##X## is that it covers ##X##. Then the collection of all unions of finite intersections of elements of ##\mathcal{S}## make up a topology ##\mathcal{T}## on ##X##. That means the collection of all finite intersections of elements of...

There are several possible topologies for an electrical circuit. However, if we limit our circuit to be a two terminal device, how will this limit the options for the different topologies? I am a beginner in this field, but as far as I can tell by drawing the circuits, the only possible...

The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...

I'm a physics major interested in taking some upper level math classes such as topology, differential geometry, and group theory but these classes are only taught in the math department and are heavy on the proofs. Analysis are recommended and preferred prerequisites but are apparently not...

So right now I am reading on continuity in topology, which is stated as a function is continuous if the open subset of the image has an open subset in the inverse image... that is not the issue. I just read the pasting lemma which states: Let X = A\cupB, where A and B are closed in X. Let...

Hi there, Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?

Assuming a mathematical plane, does it have a top and a bottom or does defining them make the plane three dimensional? Example: Given a flat, transparent plastic sheet. One draws a picture on it with a marker. If one turns the sheet over, in other words looking at the bottom of the sheet...

Hi! I'm trying to get some intuition for the notion of the topology of a set. The definition of a topology ##\tau## on a set ##X## is that ##\tau## satisfies the following: - ##X## and ##Ø## are both elements of ##\tau##. - Any union of sets in ##\tau## are also in ##\tau##. - Any finite...

Author: George F. Simmons Title: Introduction to Topology and Modern Analysis Amazon Link: https://www.amazon.com/dp/1575242389/?tag=pfamazon01-20

So, in the topology text I'm reading is mentioned that if each ##X_n## is first countable, then ##\prod_{n\in \mathbb{N}} X_n## is first countable as well under the product topology. And then it says that this does not need to be true for the box topology. But there is no justification at all...

In reading about the Tube Lemma, an example is given where the Tube Lemma fails to apply: namely, the euclidean plane constructed as R X R. The Tube Lemma does not apply here because R is not compact. The example given is as follows: Consider R × R in the product topology, that is the...

Homework Statement X is a compact metric space, X/≈ is the quotient space,where the equivalence classes are the connected components of X.Prove that X/ ≈ is metrizable and zero dimensional. Homework Equations Y is zero dimensional if it has a basis consisting of clopen (closed and open at...

Homework Statement Let X be the set of continuous functions ## f:\left [ a,b \right ] \rightarrow \mathbb{R} ##. Let d*(f,g) = ## \int_{a}^{b}\left | f(t) - g(t) \right | dt ## for f,g in X. For each f in X set, ## I(f) = \int_{a}^{b}f(t)dt ## Prove that the function ## I ##...

Homework Statement Show that if x = (x1, x2,...) and y = (y1, y2,...) are members of l^2, then \sum^{\infty}_{i=1} |x_{i}y_{i}| Converges Homework Equations My book defines l^2 to be: { x=(x_{1}, x_{2}, ... ) \in ℝ^{\omega} : \sum^{\infty}_{i=1} (x_{i})^{2} converges }...

Hi all, My Topology textbook arrived in the mail today, so I started reading it. It begins with an introduction to an object called metric spaces. It says A metric on a set X is a function d: X x X -> R that satisfies the following conditions: -some conditions-- I am not...

Would anyone have ideas on how to solve the following problem? Let (X, τ) be a Hausdorff space and τ0 = {X\K: so that K is compact in (X, τ)} Show that: 1) τ0 is a topology of X. 2) τ0 is rougher than τ (i.e. τ0 is a genuine subset of τ). 3) (X, τ0) is compact. This was a...

Hi, I am trying to prove the following proposition: Let F be a closed subset of the Euclidean space Rn.Then the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn. Any ideas?

So how does the topology of R^n minus the origin relate to that of the (n-1)-dimensional sphere? I would think the topology of the former is equivalent to that of an (n-1)-dimensional sphere with finite thickness, and open edges. But I suppose that is as close as one can get to the...

Homework Statement Let B be the family of subsets of \mathbb{R} consisting of \mathbb{R} and the subsets [n,a) := {r \in \mathbb{R} : n \leq r < a} with n \in \mathbb{Z}, a \in \mathbb{R} Show that B is not a topology on \mathbb{R} Homework Equations The Attempt at a Solution If B...

Homework Statement 1. Let A= {a,b,c}. Calculate the subspace topology on A induced by the topology T= { empty set, X,{a},{c,d},{b,c,e},{a,c,d},{a,b,c,e},{b,c,d,e},{c}, {a,c}} on X={a,b,c,d,e}.Homework Equations Given a topological space (X, T) and a subset S of X...

Whenever I try to understand deeper aspects of the higher maths involved in physics I keep hearing about topology related stuff. How useful is it to learn topology in order to get a deeper understanding on the maths behind physics? Also, what other maths should I look into? Functional analysis...

Homework Statement The Dirichlet Prime Number Theorem indicates that if a and b are relatively prime, then the arithmetic progression A_{a,b} = \{ ...,a−2b,a−b,a,a+b,a+2b,...\} contains infinitely many prime numbers. Use this result to prove that Z in the arithmetic progression topology is not...

Homework Statement Hi, This is my first post. I had a question regarding open/closed sets and subspace topology. Let A be a subset of a topological space X and give A the subspace topology. Prove that if a set C is closed then C= A intersect K for some closed subset K of X. Homework...

Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function. the graph of f is the subset ℝn × ℝk defined by G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)} with the subspace topology so I'm really just trying to understand that last part of this definition...

Homework Statement If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology? The Attempt at a Solution My understanding of discrete topology suggests to me that if A is discrete it...

Hi there, I've come across the term 'non-trivial topology' or 'non-trivial surface states' when researching topological superconductors and really need a bit of help as to exactley what this means? I've tried google but no-one seems to give a definition? Many thanks for checking this out

Hi folks ... I urgently need good books about Functional analysis and Topology. These must be comprehensive and thorough, undergraduate or graduate. Please, advise and provide your experiences with such books. I accept only thick books ;) e.g Introductory Functional Analysis with...

Metric Space and Topology HW help! Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s \ni X if \forall \epsilon > 0 \exists N \forall n ≥ N : d(sn,s) < \epsilon For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself...

Homework Statement X is a metric and E is a subspace of X (E\subsetX) The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E, ∂E=\overline{E}\cap(\overline{X\E}) (ignore the red color, i can't get it out) Show that E is open if and only...

Homework Statement Let S={zεℂ: |z|<1 or |z-2|<1}. show that S is not connected.Homework Equations My prof use this definition of disconnected set. Disconnected set - A set S \subseteqℂ is disconnected if S is a union of two disjoint sets A and A' s.t. there exists open sets B and B' with A...

Author: Bert Mendelson Title: Introduction to Topology Amazon link: https://www.amazon.com/dp/0486663523/?tag=pfamazon01-20 Level: Undergrad

Homework Statement Let X be the set of all points (x,y)\inℝ2 such that y=±1, and let M be the quotient of X by the equivalence relation generated by (x,-1)~(x,1) for all x≠0. Show that M is locally Euclidean and second-countable, but not Hausdorff. Homework Equations The Attempt at...

Author: John Milnor Title: Topology from the Differentiable Viewpoint Amazon Link: https://www.amazon.com/dp/0691048339/?tag=pfamazon01-20 Prerequisities: Level: Undergrad Table of Contents: Preface Smooth manifolds and smooth maps Tangent spaces and derivatives Regular values The...

Author: Raoul Bott, Loring Tu Title: Differential Forms in Algebraic Topology Amazon Link: https://www.amazon.com/dp/1441928154/?tag=pfamazon01-20 Prerequisities: Differential Geometry, Algebraic Topology Level: Grad Table of Contents: Introduction De Rham Theory The de Rham Complex...

Hey guys in fact here is my first time to have interaction over this forum! I've already read how one can show the topology in ℝ(real Line) which is usual called standard topology fulfill the three condition fro to be topology. however, I want to make inquiry on how can i proof whether...

Hi all, I need help with something basic but I'm not sure how to handle it. The doubt is about how to consider the topology of the unit interval I=[0,1] inherited of the real line with its usual topology (intervals of the type (a,b)). I think that is just to pay attention to the definition...

Homework Statement The textbook exercise asks for a Hausdorff topology on \{a,b,c,d,e\} which is not the discrete topology (the power set). It is from "Introduction to Topology, Pure and Applied", by Adams and Franzosa. Homework Equations Let X be a set. Definition of topology...

Let \{ [a_j, b_j]\}_{j\in J} be a set of (possibly infinitely many closed intervals in R whose intersection cannot be expressed as a disjoint union of subsets of R. Prove that \bigcup\limits_{j \in J} {\{ [{a_j},{b_j}]\} } is a closed interval in R. I don't understand how to attack this...

Author: James Munkres Title: Topology Amazon link https://www.amazon.com/dp/0131816292/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. An encounter with rigorous calculus or analysis is a plus. Level: Undergrad Table of Contents: Preface A Note...

I am new to manifolds and I read the fact that any discrete space is a 0 dimensional manifold. I am having a hard time understanding why and feel this is very basic. So to be a manifold, each point in the space should have a neighborhood about it that is homeomorphic to R^n. (and n will...

Hello, I was wondering if it was possible (or advisable) to read Chapter 7 of Munkres (Complete Metric Spaces and Function Spaces) without having done Tietze Extension Theorem, the Imbeddings of Manifolds section, the entirety of Chapter 5 (Tychonoff Theorem) and the entirety of Chapter 6...

in a nutshell, what is the difference between those two fields?

I'm a physics undergraduate and I'll starting learning topology from Munkres next semester. But first I want to learn set theory to feel more comfortable. Do you know any good textbook? A friend of mne from the math department said I should go with Kaplansky's "Set Theory and Metric Spaces".

In topology, when we say a set is closed, it means it contains all of its limit points In Algebra closure of S under * is defined as if a, b are in S then a*b is in S. Are these notations similar in any way?

Mod note: This thread contains an off-topic discussion from the thread https://www.physicsforums.com/showthread.php?p=4216768 So a notion of distance is used... I wonder how.

The question comes from the Munkres text, p. 133 #3. Let Xn be a metric space with metric dn, for n ε Z+. Part (a) defines a metric by the equation ρ(x,y)=max{d1(x,y),...,dn(x,y)}. Then, the problem askes to show that ρ is a metric for the product space X1 x ... x Xn. When I originally...