In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.
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...
Hey am new to this forum but I have a question regarding topologically protected states.. Let's suppose we have a 1D gapped system divided two to distinct regions that have different periodicity or different properties and that at the centre, where the two regions 'meet' states appear in the...
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
Prove that any simplicial complex is Hausdorff.
Homework EquationsThe Attempt at a Solution
I have proved that for any finite simplicial complex, it is metrizable and hence Hausdorff.
How to show the statement for infinite case?
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
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...
1. I have to show that
S1 = {x ∈ R2 : x1 ≥ 0,x2 ≥ 0,x1 + x2 = 2}
is a bounded set.2. So I have to show that sqrt(x1^2+x2^2)<M for all (x1,x2) in S1.3. I have said that M>0 and we have 0<=x1<=2 and 0<=x2<=2.
And x2 = 2-x1
We can fill in sqrt(x1^2 + (2-x1)^2) = sqrt (0^2 + (2-0)^2) = 2 < M = 3...
Hello everyone,
I was wondering if someone could assist me with the following problem:
Let T be the topology on R generated by the topological basis B:
B = {{0}, (a,b], [c,d)}
a < b </ 0
0 </ c < d
Compute the interior and closure of the set A:
A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3)
I...
I am having some trouble visualising the following problem and I hope someone will be able to help me:
Let (X, dx) and (Y,dy) be metric spaces and consider their product topology X x Y (T1) and the topology T2 induced by the metric d((x1,y1),(x2,y2)) = max(dx(x1,x2),dy(y1,y2)) so the maximum of...
Hi. I'm taking a look at some lectures by Charles Kane, and he uses this simple model of polyacetylene (1D chain of atoms with alternating bonds which give alternating hopping amplitudes) [view attached image].
There are two types of polyacetylene topologically inequivalent. They both give the...
I am looking at a statement that, for a short exact sequence of Abelian groups
##0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0##
if ##C## is a free abelian group then this short exact sequence is split
I cannot figured out why, can anybody help?
Homework Statement
Let ##Q = \{(x_1,x_2,...,) \in \mathbb{R}^\omega ~|~ \lim x_n = 0 \}##. I would like to show this set is closed in the uniform topology, which is generated by the metric ##\rho(x,y) = \sup d(x_i,y_i)##, where ##d## is the standard bounded metric on ##\mathbb{R}##. Homework...
Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this?
Thanks
Homework Statement
Clearly if a sequence of points ##\{x_n\}## in some space ##X## with some topology, then the sequence will also converge when ##X## is endowed with any coarser topology. I suspect this doesn't hold for endowment of ##X## with a finer topology, since a finer topology amounts...
Hi all.
I would like to ask some explanation regarding how to obtain L and C values for a butterworth filter using the first Cauer topology. I will consider a normalized low pass filter.
(Link added by Mentor)
https://en.wikipedia.org/wiki/Butterworth_filter
What I don't understand is this...
I would like to know which chapters in Munkres Topology textbook are essential for a physicist. My background in topology is limited to the topology in baby Rudin, Kreyzig's functional, and handwavy topology in intro GR books. I feel like the entire book isn't necessary, but I could be mistaken.
Hi, I'm a new member and not sure if it is ok to post a link to my blog or youtube channel. So I thought I would ask first before doing so. Is that ok or should I do it someplace else?
According to this author, http://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf, a locally Minkowski spacetime with a nontrivial global topology may have a preferred inertial frame, in the sense that hypersurfaces of constant time can only be defined using particular time...
A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C1 and C2 lie inside the third cycle C3. However C1 does not lie inside C2, nor vice-versa.
What is the configuration of the orbits?
Show that there must be at least one fixed point bounded...
I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without...
Homework Statement
Let A:={x∈ℝ2 : 1<x2+y2<2}. Is A open, closed or neither? Prove.Homework Equations
triangle inequality d(x,y)≤d(x,z)+d(z,y)
The Attempt at a Solution
First I draw a picture with Wolfram Alpha. My intuition is that the set is open.
Let (a,b)∈A arbitrarily and...
Hello,
do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)?
This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology).
Thanks in advance!
Dear Physics Forum personnel,
I will be taking my first graduate course (as an undergraduate) in mathematics starting on this Fall Semester. The course is about the algebraic topology (Hatcher, Spanier, Massey, etc.), which I am very excited to take as I love the topology. I am curious about...
I'm trying to prepare to read The Large Scale Structure Of Space-time by Hawking and Ellis. I've been reading a General Topology textbook since the authors say "While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavored to write this book so...
I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated.
----------------------
Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...
Homework Statement
Suppose that a turn of B-DNA in a circular DNA molecule with L = 100 and W = -4 becomes a turn of Z-DNA .
a) What are the L, T, and W following the transition?
Homework Equations
L=T+W
L=#bp/#bp/turn
B-dna= 10.4bp/turn
Z-dna=12bp/turn
The Attempt at a Solution...
I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the...