Topology Definition and 798 Threads

This September I will be going back to school after being away for 3-4 years. When I was going before, I took a class in point-set topology. I passed the class, but only with a 53. This wasn't for lack of ability, but for a lack of motivation. I dropped out of school after that semester and did...

Hi! I would like to know if my assumptions are right: Topology is the merging domain of analysis and algebra; Relational algebra use topological operators; Relational algebra is a specification of topology ?

Just for fun, I tried enumerating the topologies on n points, for small n. I found that if the space X consists of 1 point, there is only one topology, and for n = 2, there are four topologies, although two are "isomorphic" in some sense. For n = 3, I I found 26 topologies, of 7 types. For n...

I'm trying to get comfortable with the idea of a flat torus topology that is also an everywhere a smooth manifold like the video game screen where you got off the screen to the right and pop out on the left (because as I understand it this topology could be a model of space) I can't get how...

Following is from Wolfram Mathworld "A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: The empty set is in T.X is in T.The intersection of a finite number of sets in T is also in T.The...

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. We are going over separation axioms in class when we were asked to prove that every Urysohn Space is a Hausdorff. What I understand: A space ##X## is Urysohn space provided whenever for any...

Hello, A couple of years ago I studied abstract algebra from Dummit and Foote. However, I was never able to gain the intuition on the subject that I would like from that book. I want to study the subject again, and I want to use a different book this time around - one that covers a lot of...

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...

I am currently looking at grad schools, and I am wondering if anyone knew who are the leading researchers in differential geometry. I know that question is a little vague considering how diverse differential geometry is, but I was hoping that something could direct me in the right direction...

Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...

I am trying to understand Differential Topology using several textbooks including Lee's book on Smooth Manifolds. I am looking for some good online lecture notes at undergraduate level (especially if they have good diagrams and examples) in order to supplement the texts ... Can anyone help in...

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...

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 $T$ ?

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...

[SIZE="4"]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...

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...