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.
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...
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 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...
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'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...
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...
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...
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...
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...
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...
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...
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 ##...
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...
<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...
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...
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...
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...
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...
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...