Topology Definition and 798 Threads

Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...

Mathematicians, I summon thee to help me identify which field deals with this stuff. I come here not as a physicist but as a sunday programmer trying to solve some numerical problems. I set out to model a lattice version of a smooth space. A discretization procedure not uncommon in physics, but...

Hello there! Topological insulators and supercontuctors nowadays are very active field in physics research. I am looking for a Phd in theoretical matter physics, and these arguments could interest me. But I have a question: phisicists that study topological superconductors, insulators and...

Can a black hole be presented as a Heegaard decomposition or as the complement of a knot? I'll try and elaborate: If I understand correctly, the cross section of spacetime near a black hole can be thought of topologically as a manifold. What manifold is it? Can the manifold be decomposed?

Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...

I was asking to myself what is the usefulness of a topology. I'd thought this question before and couldn't find results on the literature, perhaps I was not searching with the right terms. So I started thinking that maybe a topology is a way of defining the domain, codomain and image of a...

Hi, I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space. Assume that we have the countable collection of dense open sets ##...

I am taking a course in topology with Gamelin and Greene, Introduction to topology. I would like to have some supplement to extend and give more motivation and explanation. I am quite tired of the "theorem, proof, theorem, proof" pattern. Thank you!

Homework Statement This is a problem from Munkres(Topology): Show that a connected metric space ##M## having having more than one point is uncountable. Homework Equations A theorem of that section of the book states: Let ##X## be a nonempty compact Hausdorff space. If no singleton in ##X## is...

Hi,t I am studying topology at the moment. I have seen that some authors define the neighborhood of a point using inclusion of an open set, while others define the term as open set that contains the point. In most of the theory I have seen so far, the latter is more convenient to use. Why is...

To start studying topology, what basic knowledge should I have?

<Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...

I want to show that ##\mathbb{R}## is disconnected with the subspace topology. For this I considered that ##\mathbb{R} = \lim_{\delta n \longrightarrow 0 } (-\infty, n] \cup [n+\delta n, \infty)## and of course the intersection of these two open sets is empty. What I'm not sure is about the...

In reading out about topological spaces and topologies I noticed that they do not give much specific examples, so I have not found an answer to the following simple question: Can we use the same function for mapping into two different open sets of a given topology? Or, perhaps equivalently, can...

Is there a way we can see why the axioms defining a topology/ topological space are the way they are?

Hello fellows My background is architecture (bachelor in2016) but for unknown reasons I’ve been fascinated by geometry since last year. it was roughly at the stage where I was trying to grasp ‘the truth ‘ of architecture and somehow got into geometry... happy coincidence. Since I hadn’t...

Hello Forum, Does topology reckon the art of turning a square into a circle? I am quite new to topology and maths in general, I have only dabbled and eyed on my collection of mathbooks. I have come to a conclusion of how to turn the Square into A Circle without cutting. I wonder if I am...

Homework Statement Let ##X=([1,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## and ##Y=((0,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## ##a)##Find subspaces of of the euclidean plane ##\mathbb{R}^2## which are homeomorphic to the compactification with one...

There are couple things that keep me questioning about the nature of the universe. Let me start from the begining. Big Bang happened and our universe was created, and from now on let us suppose that the universe is infinite in size. Later on, the universe expands and after a time we can see...

For anyone who is familiar with the book "Geometry, Topology and Physics" by Nakahara, what do you think are the mathematical and physics prerequisites for this book ?

Homework Statement We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this ##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## ##a)## Show that ##\tau## satisfies that axioms for...

Homework Statement show that the two topological spaces are homeomorphic. Homework Equations Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them The Attempt at a Solution I have tried proving that these two spaces are homeomorphic...

Homework Statement There was the times 100 years ago, N.Herreshoff was designing giant J Boats, America s Cup boats by only carving a wood piece in few hours ,without drawing calculating anything and builders were measuring the wooden half model and building a multi million dollar yacht wins...

I am encountering this kind of problem in physics. The problem is like this: Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between...

So according to Dr. Frederic Schuller, we need to at least know the topology on the space of all theories in order to know that we are getting closer to the truth. I take that this is because we need to know the topology to establish that convergence is possible in the first place. How does this...

I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...

Holy bleeping bleepity bleep bleep! I now will have a Master's in mathematics - and a lot of extra time (for my family of course). -Dave K

Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...

In classical general relativity, gravity is simply a curvature of space-time. But, a quantum field theory for a massless spin-2 graviton has as its classical limit, general relativity. My question is about the topology of space-time in the hypothetical quantum field theory of a massless spin-2...

Eugene Wigner once famously talked about the "unreasonable effectiveness of mathematics" in describing the natural world. Today again we are seeing this in action in particular with regard to the description of the biological brain from the perspective of neuroscience. Researchers from the Blue...

Hi, Regarding transmission line booster transformer topologies I'm curious as to what would happen over all the possible permutations. I refer you to the following: http://top10electrical.blogspot.com.au/2015/03/booster-transformer.html I presume that in real life the booster transformers...

Advanced Physics (Advanced Science) by Steve Adams & Jonathan Allday from OUP Oxford: and Physics (Collins Advanced Science) 3rd Edition by Kenneth Dobson from Collins Educational: http://www.amazon.com/dp/0007267495/?tag=pfamazon01-20 Does anyone know any of these books? I find them very...

Hello, I have read a fair chunk of Munkres' Topology book and took a short introductory course during undergraduate, but I would like to learn point-set topology a little better. I have quite a bit of mathematical maturity, so that isn't an issue for me. I had a larger list of potential books to...

Homework Statement [/B] Is {n} an open set?Homework Equations [/B] To use an example, for any n that is an integer, is {10} an open set, closet set, or neither?The Attempt at a Solution [/B] I say {10} is a closed set, because it has upper and lower bounds right at 10; in other words, it is...

How much of topology one needs to know to have a great knowledge of the math of Special and General Relativity? I'm asking this because I'm interested in really look at the theory of Relativity with the eyes of a mathematician. I suppose that just knowing what a manifold is or even what a...

Homework Statement (part of a bigger question) For ##x,y \in \mathbb{R}^n##, write ##x \sim y \iff## there exists ##M \in GL(n,\mathbb{R})## such that ##x=My##. Show that the quotient space ##\mathbb{R}\small/ \sim## consists of two elements. Homework EquationsThe Attempt at a Solution Well...

Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate...

Hello! I just started reading an introductory book about topology and I got a bit confused from the definition. One of the condition for a topological space is that if ##\tau## is a collection of subsets of X, we have {##U_\alpha | \alpha \in I##} implies ##\cup_{\alpha \in I} U_\alpha \in \tau...

Homework Statement Let ##X## and ## Y## be non-empty sets, ##i## be the identity mapping, and ##f## a mapping of ##X## into ##Y##. Show the following a) ##f## is one-to-one ##~\Leftrightarrow~## there exists a mapping ##g## of ##Y## into ##X## such that ##gf=i_X## b) ##f## is onto...

Homework Statement If ##\bf{A}## ##= \{A_i\}## and ##\bf{B}## ##= \{B_j\}## are two classes of sets such that ##\bf{A} \subseteq \bf{B}##, show that ##\cap_j B_j \subseteq \cap_i A_i## and ##\cup_i A_i \subseteq \cup_j B_j## Homework EquationsThe Attempt at a Solution Since ##\bf{A} \subseteq...

What is the purpose of the shim inductor and how exactly does it function in PSFB topology? I noticed that I had dramatically improved efficiency when I added a shim inductor to my circuit, not too sure how or why this works though.

Hi, I would like to receive suggestions regarding (general) topology textbook for self-study. I have background in real analysis, linear and abstract algebra. I am not afraid of a challenging book. Thank you!

Homework Statement Let ##A## be a subspace of a topological space ##X##, and let ##B\subset A##. Show that ##B## is closed if and only if there exists a closed subset ##C \subset X## such that ##B = C \cap A## Homework EquationsThe Attempt at a Solution So I've started by just drawing the...

Homework Statement ##X = \{1,2,3\}## , ##\sigma = \big\{\emptyset , \{1,2\}, \{1,2,3\} \big\}##, topology ##\{X, \sigma\}## ##Y = \{4,5\}## , ##\tau = \big\{\emptyset , \{4\}, \{4,5\} \big\}##, topology ##\{Y, \tau\}## ##Z = \{2,3\} \subset X## Find all the open sets in the subspace topology...

Homework Statement See attached picture.Homework EquationsThe Attempt at a Solution At the moment, I am dealing with part (a). What I am perplexed by is the ordering of the parts. If the subbasis in part (b) does indeed generate this coarsest topology, why wouldn't showing this be included in...

Homework Statement Determine whether the following subsets are open in the standard topology: a) ##(0,1)## b) ##[0,1)## c) ##(0,\infty)## d) ##\{x \in (0,1) : \forall n \in \mathbb{Z}^{+}## ##, x \not= \frac{1}{n}\} ## Homework EquationsThe Attempt at a Solution a) ##(0,1)## is open because...

Homework Statement I am trying to show that there exists a metric on ##\mathbb{R}^2## that induces the dictionary order topology on the plane. Homework EquationsThe Attempt at a Solution If I recall correctly, vertical intervals in the plane form basis elements for the dictionary order...

Homework Statement I have a set I = {x from R3 : x1<1 v x1>3 v x2<0 x x3>-1} Homework Equations Open disc B (x,r) (sqrt (x-x0)^2 + (y-y0)^2) < r The Attempt at a Solution I have done, for example by x1<1, that let r = 1-x1 Then sqrt ((x-x1)^2 + (y-y1)^2) < sqrt (x-x1)^2) < r = 1-x1 So |x-x1|...

Hello, I have a practical problem, I'd like to find the "best" spot to hear sounds in a valley (forgive me if "acoustic point" isn't an appropriate term, I just couldn't come up with anything better and scrolling an acoustics text didn't help), or at least a non-blind spot (one which instead...