My Math Forum Doubts on topology

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February 27th, 2017, 07:10 PM   #21
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 Originally Posted by Lalitha183 I have taken courses on Algebra, Real analysis and vectors, group theory ring theory. You mean proof writing theorems or a course which teaches how to write a proof? I have been into proofs and theorems whole year in 2nd year of my graduation and havent taken any math course that teaches how to write a proof.
Ok real analysis is good. Do you know what an open set in the real numbers is?

If so, can you prove that the union of open sets of reals is open, and that a finite intersection is open, and that the empty set is open, and that the reals are open as a subset of the reals? That's the starting point for general topology, since those are exactly the properties of open sets that we are going to abstract.

 February 27th, 2017, 07:13 PM #22 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 153 Thanks: 1 Can you please help me in understanding every topic clearly. Can you suggest me books or videos or anything that gives me basic understanding of the concept before moving into problems. I was not able to attach the pdf here. so here it is : CMI M.SC (MATHS) ENTRANCE EXAM SYLLABUS Important note The syllabus includes topics for PhD entrants too and so contains material which may often be found only in MSc courses and not BSc courses in the country. Our policy generally has been to have a common question paper for MSc and PhD levels but have separate cut-offs for them. The Syllabus Algebra. (a) Groups, homomorphisms, cosets, Lagrange’s Theorem, group actions, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields, algebraic extensions, finite fields (b) Matrices, determinants, vector spaces, linear transformations, span, linear independence, basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigenvectors, upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rota- tions, orthogonal matrices, GLn, SLn, On, SO2, SO3. References: (i) Algebra, M. Artin (ii) Topics in Algebra, Herstein (iii) Basic Algebra, Jacobson (iv) Abstract Algebra, Dummit and Foote Complex Analysis. Holomorphic functions, Cauchy-Riemann equations, integration, zeroes of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville’s theorem, poles and sin- gularities, residues and contour integration, conformal maps, Rouche’s theorem, Morera’s theorem References: (i) Functions of one complex variable, John Conway (ii) Complex Analysis, L V Ahlfors (iii) Complex Analysis, J Bak and D J Newman Calculus and Real Analysis. (a) Real Line: Limits, continuity, differentiablity, Reimann integration, sequences, series, lim- sup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions, (b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives, total derivative, Jacobian, gradient, line integrals, surface integrals, vector fields, curl, di- vergence, Stoke’s theorem (c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation. References: (i) Principles of mathematical analysis, Rudin (ii) Real Analysis, Royden (iii) Calculus, Apostol Topology. Topological spaces, base of open sets, product topology, accumulation points, bound- ary, continuity, connectedness, path connectedness, compactness, Hausdorff spaces, normal spaces, Urysohn’s lemma, Tietze extension, Tychonoff’s theorem, References: Topology, James Munkres Last edited by Lalitha183; February 27th, 2017 at 07:19 PM.
February 27th, 2017, 07:14 PM   #23
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 Originally Posted by Maschke Ok real analysis is good. Do you know what an open set in the real numbers is? If so, can you prove that the union of open sets of reals is open, and that a finite intersection is open, and that the empty set is open, and that the reals are open as a subset of the reals? That's the starting point for general topology, since those are exactly the properties of open sets that we are going to abstract.
ok. I l try to prove them and will reply you back.

February 27th, 2017, 07:25 PM   #24
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 Originally Posted by Lalitha183 ok. I l try to prove them and will reply you back.
What is your definition of an open set of real numbers?

February 27th, 2017, 07:27 PM   #25
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 Originally Posted by Maschke What is your definition of an open set of real numbers?
A non empty set whose limit points are not defined in the set is an open set.

February 27th, 2017, 07:34 PM   #26
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 Originally Posted by Lalitha183 A non empty set whose limit points are not defined in the set is an open set.
* Is the empty set open?

* What is a limit point?

* Does the set of real numbers contain all its limit points? Yes it does, right? But isn't it an open set?

I hope I don't seem like I'm being picky for the sake of being picky. I'm being picky because we have to get to something that we can use as a basis to work our way back up to topology. We're not there yet.

ps -- Have a look at this. Read about the definition of an open set in Euclidean space, a metric space, and the definition of a topological space. Does this make sense? You will find it helpful to spend some time with this article till it becomes familiar.

https://en.wikipedia.org/wiki/Open_set#Euclidean_space

Last edited by Maschke; February 27th, 2017 at 07:44 PM.

February 27th, 2017, 07:40 PM   #27
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Quote:
 Originally Posted by Maschke * Is the empty set open? * What is a limit point? * Does the set of real numbers contain all its limit points? Yes it does, right? But isn't it an open set? I hope I don't seem like I'm being picky for the sake of being picky. I'm being picky because we have to get to something that we can use as a basis to work our way back up to topology. We're not there yet.
As an empty set contains no elements except $\pi$ ( here I mean pi which shows a null set. I dont know how to represent that symbol here) which is a subset of every subset of Reals it is open.

Limit points define the boundaries of the set which defines the elements contained in it.

Set of Reals $R$ contains infinitely many points so I guess it doesnt contain its limit point so it could be open

February 27th, 2017, 08:06 PM   #28
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 Originally Posted by Lalitha183 As an empty set contains no elements except $\pi$ ( here I mean pi which shows a null set. I dont know how to represent that symbol here) which is a subset of every subset of Reals it is open.
$\emptyset$. Markup is \emptyset. But what do you mean it contains $\pi$? the empty set contains no elements at all, right?

Quote:
 Originally Posted by Lalitha183 Limit points define the boundaries of the set which defines the elements contained in it.
Do you still have your real analysis text? Can you refer to it and find where they define a limit point of a set of real numbers? Can you write down the definition? For that matter, before you begin to study topology you must master the definitions of open set, limit, continuity, etc. directly from your real analysis text.

Quote:
 Originally Posted by Lalitha183 Set of Reals $R$ contains infinitely many points so I guess it doesnt contain its limit point so it could be open
The set of integers as a subset of the reals is infinite, it is open? The closed unit interval $[0,1]$ is infinite and it contains its limit points. It's closed and not open.

Do you have time to review your real analysis? Just the definition of open set, limit point, neighborhood, ball, interior point. If you understand interior points in the real numbers then they're very easy in topology.

I am sorry we need to go backward before going forward but one we understand open sets in the real numbers general topology is very simple.

Depending on your timeframe, if you have $X$ weeks before your topology test you should spend $X/2$ weeks reviewing real analysis. If you really understand real analysis the topology stuff is really easy, you just breeze right through it because it's intuitive and obvious.

But if you are hazy on what is an open set or a limit point in the reals, the topological definitions will not make sense because you have nothing to relate them to.

Last edited by Maschke; February 27th, 2017 at 08:10 PM.

February 27th, 2017, 08:08 PM   #29
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Quote:
 Originally Posted by Maschke $\emptyset$. Markup is \emptyset. But what do you mean it contains $\pi$? the empty set contains no elements at all, right? Do you still have your real analysis text? Can you refer to it and find where they define a limit point of a set of real numbers? Can you write down the definition? For that matter, before you begin to study topology you must master the definitions of open set, limit, continuity, etc. directly from your real analysis text. The set of integers as a subset of the reals is infinite, it is open? The closed unit interval $[0,1]$ is infinite and it contains its limit points. It's closed and not open. Do you have time to review your real analysis? Just the definition of open set, limit point, neighborhood, ball, interior point. If you understand interior points in the real numbers then they're very easy in topology. I am sorry we need to go backward before going forward but one we understand open sets in the real numbers general topology is very simple.
Ok I will refer to it

February 27th, 2017, 08:12 PM   #30
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 Originally Posted by Lalitha183 Ok I will refer to it
I promise we'll be at the bottom of this soon then we can go forward.

We need to know in the real numbers: What is an open set, what is a limit point, what is an interior point, what is a boundary point, what is a closure, and what is a closed set. That list of terms. And we need exact definitions. Once we have those things we can prove that the open sets in the reals are a topology, and from then on whenever we're doing topology we just think about the reals. That gets you through 90% of topology at this level.

I can give you some definitions but I'd rather work from definitions in a book that you own.

Ok here is the official list:

* open set

* limit point

* interior point

* boundary point

* closure of a set

* interior of a set

* closed set

* open ball

* neighborhood

If you look up those terms and write down the exact definitions from your book we can go forward from there. I'm trying to avoid having to develop this material on my own because my presentation will inevitably conflict with whatever's in your book and it will cause confusion. Let's work from your real analysis book. Which book is that by the way? If I can find a pdf of it online I can see if I can help you find these defs.

Remember we are not actually going backward. Knowing those definitions is pretty much all you need to understand elementary general topology.

Last edited by Maschke; February 27th, 2017 at 08:19 PM.

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