Topology Definition and 818 Threads

For a space (X,T) must there be a topology W on X coarser than T such that (X,W) is semiregular other than the indiscrete topology and if so are there two such nonhomeomorphic topologies neither of which are the indiscrete topology? I know that any regular space is semi regular and that for...

In Munkres book "Topology" (Second Edition), Munkres proves that a function F is a homeomorphism ... I need help in determining how to find the inverse of F ... so that I feel I have a full understanding of all aspects of the example ... Example 5 reads as follows:Wishing to understand all...

The title above give my name. I am a pure maths PhD with an interest in physics and geometry. I am currently studying physics for fun and I am very interested in current progress. I am especially interested in quantisation of space time, holographic theories and dualities. Regards John

It is said that the metric tensor in GR is generally covariant and obey diffeomorphism invariance.. but the signature, boundary conditions and topology are not. What would be GR like if these 3 obey GC and DI too? Is it possible?

I am working on this problem on measure theory like this: Suppose ##X## is the set of real numbers, ##\mathcal B## is the Borel ##\sigma##-algebra, and ##m## and ##n## are two measures on ##(X, \mathcal B)## such that ##m((a, b))=n((a, b))< \infty## whenever ##−\infty<a<b<\infty##. Prove that...

Hi, I'm hoping someone here can shed some light, I'm currently in my 3rd year of my Physics degree and have discovered I really don't have the mind to memorise / reproduce paragraphs of text. Even if I understand the concepts it takes me a LONG time for my brain to take text in. Maths however I...

Hey guys, long story short. I am completing my Math minor this semester and need to decide on whether Topology or Fourier Analysis. I am an undergraduate physics major and neither one of those classes is required for my B.S. in physics. So what do you guys think, Topology or Fourier Analysis?

This might be well known or even discussed here, though I couldn't find a thread about it, but the questions is what are the possible topologies of a black hole i.e. the topology of a spatial slice of the event horizon. I know there is a result of Hawking that says the topology has to be that of...

Hello, I know that given a set $X$ and a topology $T$ on $X$ that a basis $B$ for $T$ is a collection of open sets of $T$ such that every open set of $T$ is the Union of sets in $B$. My question is: does taking the set of all Unions of sets in $B$ give exactly the topology $T$ ?

I have been looking through the AMS and NSF websites for REUs. And I have found some in particular that I am interested in(UChicago, Cornell, SMALL, MAPS-REU), but it seems these are the only ones that have projects that deal with topology/analysis. Do any of you have any suggestions for REUs...

I guess the usual answer would be to learn as much as possible. Some background about me: I am not a physicist but I'd like to pursue a PhD in theoretical physics (after a year or two) and work on topological quantum computing. I am familiar with quantum mechanics and solid state physics (at...

So, I was trying to do a derivation of my own for the FLRW metric, since I couldn't understand the one Wald had. The spatial slice M is a connected Riemannian manifold which is everywhere isotropic. That is, in every point p\in M and unit vectors in v_1,v_2\in T_p\left(M\right) there is an...

In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...

Hi, What are the convergent sequences in the box topology on R^ω?Are they the eventually constant only? Thank's in advance

I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal representation of the universe: "Neither does the theory of relativity, Bohr argued, provide us...

Homework Statement Use technique of completing squares to Show that this function has an absolute minimum. f(x, y) = x^2 + y^2 − 2x + 4y + 1 Homework Equations Not entirely sure how completing the squares will indicate an absolute minimum.Is there some additional reasoning required? The...

My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this...

So i just recently had to drop two math courses, topology, math logic, because my math maturity wasn't up to the level needed to excel in them. I intend on taking them again, but not without first more preparation which leads to my question. Which order would i benefit more from in preparing for...

I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization ϕ^{-1}_i:π(U_i)→U_i×G . For example ϕ^{-1}_i:π(S^2)→S^2×U(1) (if that makes sense) . But I don't know how to think of this (and other products with lie groups like that)...

Let n <= m and G:=Gr(n,m) be the (real) Grassmanian manifold. I understand the topology of the simplest case, that of projective space, and am wondering if there is a way to interpret the topology of the G to similar to projective space, with the according generalizations needed. If V^n is an...

Consider the maps h: R^w (omega) ---> R^w (omega) , h (x1, x2, x3,...) = (x1,4x2, 9x3,...) g: (same dimension mapping) , g (t) = (t, t, t, t, t,...) Is h continuous whn given the product topology, box topology, uniform topology? For the life of me i am...

Hi, I'm currently studying the topology of the cuk converter and I'm wondering why do you hhave to add that first inductor to the topology? Can't you just charge the capacitor straight through the voltage source? Thank you.

Hello, I learned that there are 4 types of approach to topology: (1) General (2) Algebraic (3) Differential (4) Geometrical To have a rough understanding of General relativity, which branch of topology should I study? Thanks.

Assume ##|X| > \rho## , let ##r = |X| - \rho## Now I am trying to show that ##B(r,x)\subseteq S^c## This should be a simple question, but I am struggling trying to find the right inequlity. Attempt: let ##y## be a point in ##B(r,x)##. I know that ##|x - y| < r##. I have to somehow show...

Homework Statement . Let ##X## be a set and ##\mathcal A \subset \mathcal P(X)##. Prove that there is a topology ##σ(A)## on ##X## that satisfies (i) every element of ##A## is open for ##σ(A)## (ii) if ##\tau## is a topology on ##X## such that every element of ##\mathcal A## is open for...

Hi all. I started a thread a while back about RF mixer design. I didn't know what to do or what design to choose. You guys laid some options for me and after some research and time I have finally decided that I will go for a double balanced, ring diode topology. Here is a schematic from google...

Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...

Definition/Summary A Topological Space can be defined as a non empty set X along with a class of sets, called a topology on X which is closed under (1) arbitrary unions (2) finite intersections. It can be assumed that this will always include X and p (the empty set), but understanding the...

I was suprised to realize that foliation theory was actually closely related to topology. Indeed, http://www.map.mpim-bonn.mpg.de/Foliations, states a theorem which say that a codimension one foliation exists if and only if the Euler characteristic of the topological space is one! I am...

The intuitive picture I have of giving a set a topology, is that of giving it a shape in the sense of connecting the points and determining what points lie next to each other (continuity), the numbers of holes of the shape, and what parts of it are connected to what. However, the most...

I've been thinking about this for around 7 months now, which is way too much; Munkres seems like the "typical" introduction to topology book (kind of like how Griffiths is the "typical" E&M text), and various people (from the reviews over at Amazon) make it seem like it is an undergraduate level...

I am reading myself into a basic understanding go topology with a view to algebraic topology. I try to get a visual picture and intuitive feel for what I am learning .. but wonder if I am worrying too much about gaining this type of understanding early on ... For example I am at the moment...

I am reading Martin Crossley's book, Essential Topology. I am at present studying Example 5.55 regarding the Mobius Band as a quotient topology. Example 5.55 Is related to Examples 5.53 and 5.54. So I now present these Examples as follows: I cannot follow the relation (x,y) \sim (x', y')...

Example 1 in James Munkres' book, Topology (2nd Edition) reads as follows: Munkres states that the map p is 'readily seen' to be surjective, continuous and closed. My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed. Regarding the...

I have a basic (very basic :)) understanding of the elements of algebra and many years ago I did a course in analysis ... and I would very much like to read my way to an understanding of algebraic topology .. I figured I should start with some basic texts on topology that (hopefully) head...

I'm a phyiscs student and I have been looking at these lectures: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8 But I have never learned anything about topology before and was he covers doesn't look like the Topology chapter in my mathematical physics book. I was looking for...

Homework Statement For a metric space (X,d) and a subset E of X, define each of the terms: (i) the ball B(p,r), where pεX and r > 0 (ii) p is an interior point of E (iii) p is a limit point of E Homework Equations The Attempt at a Solution i) Br(p) = {xεX: d(x.p)≤r}...

I have recently been assigned a project in my undergraduate topology class. I would like to do something in physics which involves topology, but I am having trouble finding a basic topic. I understand that there are some very advanced topics in string theory and the like, but I would like to...

contradictory set of abelian groups.

The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...

Hello, This thread is about the two books by Naber: https://www.amazon.com/dp/1461426820/?tag=pfamazon01-20 https://www.amazon.com/dp/0387989471/?tag=pfamazon01-20 The topics in this book seem excellent. They are standard mathematical topics such as homotopy, homology, bundles...

Homework Statement Suppose we have f: \mathbb C^n \to \mathbb C^n with f(z_1, \cdots,z_n) = (\sigma_1(\mathbf z), \cdots, \sigma_n(\mathbf z)) where the sigmas are the elementary symmetric functions (i.e. \sigma_1 = \sum z_j \quad \sigma_2 = \sum_{i < j} z_i z_j \quad etc) Note if we look...

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...