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.
I would like to study the path components (isotopy classes) of the diffeomorphism group of some compact Riemann surface. To make sense of path connectedness, I require a notion of continuity; hence, I require a notion of an open set of diffeomorphisms. What sort of topology should I put on the...
If τ is the co-finite topology on an infinite set X, does there exist an injection from τ to X? I'm having trouble wrapping my mind around this.
on the one hand, for A in τ, we have A = X - S, for some finite set S. so it seems that there is a 1-1 correspondence:
A <--> S, of τ with the...
[topology] "The metric topology is the coarsest that makes the metric continuous"
Homework Statement
Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that d: X \times X \to \mathbb R is continuous (for the product topology on X...
Hello,
A bit of background, I intend to major in physics and mathematics, and I am currently in second year. As it stands at the moment I am only enrolled in three units, and I was wondering If I should do, normally a third year unit, Introduction to Geometric Topology, (i can apply for an...
Homework Statement
It might not be a real topology question, but it's an exercise question in the topology course I'm taking. The question is not too hard, but I'm mainly doubting about the terminology:
Homework Equations
N.A.
The Attempt at a Solution
I would think not, unless I'm...
Topology Proof: AcBcX, B closed --> A'cB'
Homework Statement
Prove:
AcBcX, B closed --> A'cB'
and where the prime denotes the set of limit points in that set
X\B is the set difference
Homework Equations
Theorem:
B is closed <--> For all b in X\B, there exists a neighborhood U...
I've came across a book about topolgy, Topology Without Tears by Sidney A. Morris. It can be found here: http://uob-community.ballarat.edu.au/~smorris/topology.htm
The explanations are rather clear and an outline of the proof is given before each proof. However, many quite important concepts...
Homework Statement
X - topological compact space
\Delta = \{(x, y) \in X \times X: x=y \} \subset X \times X
\Delta = \bigcap_{n=1}^{\infty} G_{n}, where G_{1}, G_{2}, ... \subset X \times X are open subsets.
Show that the topology of X has a countable base.
Homework Equations
The Attempt...
Homework Statement
Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A\inTx, B\inTy } a topology on X × Y?
The attempt at a solution
I know that in order to prove T is a topology on X × Y I need to prove:
i. (∅, ∅)\inT and (X × Y)\inT
ii. T is closed under...
Just to make sure that I'm not overlooking anything, is the following an example of a quotient map p: X \to Y with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), \forall y \in Y: p^{-1}(\{ y \}) \subset X also pathwise connected and...
Homework Statement
Let X be a compact and locally connected topological space. Prove that by identifying a finite number of points of X, one gets a topological space Y that is connected for the quotient topology.
Homework Equations
The components of a locally connected space are open...
Hello
i studied Sadri Hassani az mathematical physics book.
if i want to learn topology (( for general relativity )) what it the best book for introduction ?
I'm pursuing dual degrees in mathematics and computer science with a concentration in scientific computing and am trying to decide whether I should take intro to topology or number theory.
Interests in no order are computational complexity, P=NP?, physics engines, graphics engines...
Topology of charm decays. Help!:)
Hello Everyone:)
It's my first post ever and I'm asking for help, sorry!
I have to know how to identify charm decays in the films of the Na27 experiments, done in the 80's. It was used a bubble chamber and a spectrometer...
In the paper it's said that...
Hello,
Just out of curiosity, where would following "seperation axiom" fit in?
So far I'm only acquainted with the T1, T2, T3 and T4 axioms (and the notion of completely regular in relation to the Urysohn theorem).
Hey guys,
I want to study algebraic topology on my own. I just finished a semester of pointset topology and three weeks of algebraic topology. We did not use a textbook. Can anyone recommend a book on algebraic topology?
Hatcher is fine but it is not as rigorous as I want. Munkres has...
Homework Statement
I've been given the following problem:
"Suppose that U is a finite-dimensional subspace of a Fréchet space (V,\tau). Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V."
I feel a bit...
Hi! I have this two related questions:
(1) I was thinking that \mathbb{Q} as a subset of \mathbb{R} is a closed set (all its points are boundary points).
But when I think of \mathbb{Q} not like a subset, but like a topological space (with the inherited subspace topology), are all it's...
I have a friend who, like me, is a Math major, although she started later than I did and as such, hasn't yet gotten into the core classes for her degree. She's frequently checked out my own personal library and I figured that, since the holidays are coming up, it might be cool to start her off...
Does anyone know what \Lambda means in the collection of elements:\{ A_{\lambda} : \lambda \in \Lambda\}
For example in the definition of a topology: if \{ A_{\lambda} : \lambda \in \Lambda\} is a collection of elements of a topology then \bigcup _{\lambda \in \Lambda} A_{\lambda} is in the...
Hi.
I wanted to know in what way the group of translations on a real line with discrete topology (let's call it Td) will be different from the group of translations on a real line with the usual topology (lets call it Tu)? Is Td a Lie Group? Will it have the same generator as Tu?
Hi!
I'm a beginner for a subject "topology".
While studying it, I found a confusing concept.
It makes me crazy..
I try to explain about it to you.
For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function.
I've also learned that for a set...
Homework Statement
I need two counter examples, that show the following two theorems [B]don't/B] hold:
Let X be a topological space.
1. If from the closeness of any subset A in X follows compactness of A, then X is compact.
2. If from the compactness of a subset A in X follows closeness...
Hi, I am trying to decide whether I should take a modern algebra or topology course next semester. I have a bachelor's in physics but I have not taken very many higher math classes. This is a list of the relevant classes I have taken.
Calculus (up through partial differential equations)...
Hi. I did my undergraduate work in mechanical engineering and I am working on a PhD in fluid mechanics right now. I am interested in expanding my mathematical toolbox to include topology and am looking for some advice on where to start.
What subjects/topics should I cover as a prerequisite to...
Hi,
I want to study differential topology by myself,
and i am looking for a clear book that emphesizes also the intuitive aspect.
I will be grateful to get some recommendations.
Thank's
Hedi
I must say thusfar I read through chapter one of May's book and chapter 0 of Hatcher's, May is much more clear than Hatcher, I don't understand how people can recommend Hatcher's text.
May is precise with his definitions, and Hatcher's writes in illustrative manner which is not mathematical...
Hi, the problem I am referencing is section 33 problem 4.
Let X be normal. There exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A, if and only if A is a closed G(delta) set in X.
My question is about the <= direction.
So let B be the...
Homework Statement
E is a compact set, F is a closed set. Prove that intersection of E and F is compact
Homework Equations
The Attempt at a Solution
On Hausdoff space (the most general space I can work this out), compact set is closed. So E is closed. So intersection of E and F is...
Homework Statement
The true problem is too complicated to present here, but hopefully somebody can give me a hand with this simplified version. Consider the set H = \{ (x,y) \in \mathbb R^2 : y \geq 0 \} . Denote by \partial H = \{ (x,0) \}. Let U and V be open sets (relative to H) such that...
Hi, I am enrolled in an Msc programme in pure maths, I wanted to ask for your recommendations on taking a basic graduate course in Algebraic Topology.
Basically my interest spans on stuff that is somehow related to analysis, geometry or analytic number theory.
The pros for choosing this...
Hi all!
I haven't posted here in some time, and I am in need of the expertise of you fine folks. I am busy doing some work on spin geometry. Now, as you guys know, spin structures exist on manifolds if their second Stiefel-Whitney class vanishes. This class is an element of the second...
Not sure if this is the correct place to post this, please move if need be.
I am currently learning about limit points in my Topology class and am a bit confused. Going by this:
As another example, let X = {a,b,c,d,e} with topology T = {empty set, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}. Let...
[Topology] why the words "finer" and "coarser"?
Hello,
I'm following an introductory course on topology.
Why is it that a topology with lots of opens is called fine, and one with a few ones is called coarse?
More specifically: why is this terminology more logical than the reverse (i.e...
Trying to prove:
The usual topology is the smallest topology for R containing Tl and Tu.
NOTE: for e>0
The usual topology: TR(R)={A<R|a in A =>(a-e,a+e)<A}
The lower topology: Tl(R)={A<R|a in A =>(-∞ ,a+e)<A}
The upper topology: Tu(R)={A<R|a in A =>(a-e, ∞)<A}
3. The Attempt at a...
If a point particle is a string viewed end on in three dimensions, or is a zero brane, isn't this merely a perspective? Are open strings really more like ribbons given the dimension of time and are closed strings more like tubes seen end on? If so, could this mean that the string's length is...
Hi, All:
I am trying to understand better the similarity between the compactness theorem
in logic--every first-order sentence is satisfiable (has a model) iff every finite subset
of sentences is satisfiable, and the property of compactness : a topological space X is
said...
I am a Mathematics and Physics double major, currently in my second year. I really enjoy both subjects, but my interests are progressing towards Theoretical physics/mathematical physics. My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a...
I'm making this thread because in a few weeks I'll be starting with teaching a topology course. I think I did pretty well last time I teached it, but I want to do some new things. The problem with the system in our country is that we hardly assign problems that students should solve. Students...
Can I simply combine the unions into a single interval like I did in (a)? The closed interval [1,4] fills in the (3) hole from (0,3), etc. I did something similar in (b).
http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110906_221620.jpg...
I know there are some threads out there already, but none really help me (see my description below).
I am a high school student. My highest level of math education is Calculus I. I am currently taking Calculus II (although I already know the integration portion of this course).
I have no...
Homework Statement
This is from Lee's Introduction to Smooth Manifolds. Suppose π : X → Y is a quotient map. Prove that the restriction of π to any saturated open or closed subset of X is a quotient map.
Homework Equations
Lee defines a subset U of X to be saturated if U = π-1(π(U)). π...
So I need to decide by tomorrow, whether I'll be taking topology or diff geo, (along with real analysis and advanced linear algebra). I've sat in on both classes for the first lecture, and I'm still not certain which class would be more difficult. My diff geo class has no exams, and instead...
In at least one book and one Wikipedia article, I've seen someone specify which sequences are to be considered convergent, and what their limits are, and then claim that this specification defines a topology. I'm assuming that this is a standard way to define a topology. I want to make sure that...
Background: I'm a computer science major, but who has done a lot of math (real analysis, linear/abstract algebra, combinatorics, probab&stats, numerical analysis, linear programming) and currently doing undergraduate research in computational algebra/geometry.
I'm taking a graduate level...
Definitions of "topology" and "analysis"
How do you define "topology" and "analysis"? I'm tempted to say that topology is the mathematics of...anything that involves limits. (Open and closed sets, continuous functions, etc...they can all be defined in terms of limits). But if that's an...
Can anyone please help me with this because I'm really getting confused. Thanks!
In R, we know that fine topology is equivalent to the Euclidean topology as convex functions are continuous.
Now if instead of R we consider a subset of it say [0,1], the fine topology induced on [0,1] would...
Background:
I'm going to be a junior, having very strong Analysis and Algebra yearlong sequences, in addition to a very intense Topology class, and a graduate Dynamical Systems class.
For this coming Fall, I'm sort-of registered for this class, titled "Manifolds and Topology I" (part of a...
And what's considered modern mathematics? I always thought it was 1960s+. Around 50 years ago till now is what i considered modern math.
Anyway, how important is topology? I've heard people say "the idea of evolution to biology is the same as the ideas of topology to mathematics." So is it...