General topology Definition and 26 Threads

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness:

Continuous functions, intuitively, take nearby points to nearby points.
Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
Connected sets are sets that cannot be divided into two pieces that are far apart.The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

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  1. K

    A Alternative topology of our 3D space

    Mathematically, a flat 2D surface folded down over the edge of a desk has no curvature, so a Flatlander could not detect the fold. Same for us in space if our 3D space was folded down in a 4 dimensional sense. Suppose 3D space was folded down in two different perpendicular directions, both...
  2. caffeinemachine

    I New Video Course on General Topology on YouTube

    Hello Everyone! I created a YouTube channel (here's the link) a few years ago in which I post detailed lectures in mathematics. I just started a series on General Topology. Following is a snapshot from a video. I mean to deliver a comprehensive course with a lot of pictures and intuition and...
  3. L

    I How do I use the four axioms of a neighborhood to define an open set?

    How do I define an open set using only the four axioms of topological neighborhoods, as per the Wikipedia article on topological spaces? The intuitive definition of an open set is that it's a set of points on a real number line containing only points at which there is room for some hypothetical...
  4. L

    I How to define an open set using the four axioms of a neighborhood

    I am struggling to define an open set using the four axioms of a topological neighborhood, as per the Wikipedia article "Topological spaces." An open set on a real number line is a set of points that contains only interior points, meaning that there is always room for some hypothetical particle...
  5. V

    A Metric of a Moving 3D Hypersurface along the 4th Dimension

    Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##. Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
  6. patric44

    T_{0} - Space Equivalent Definition

    Hello everyone, Concerning the separation axioms in topology. Our topology professor introduced the equivalent definition for a topological space to be a ##T_{o}-space## as: $$ (X,\tau)\ is\ a\ T_{o}-space\ iff\ \forall\ x\ \in X,\ \{x\}^{\prime}\ is\ a\ union\ of\ closed\ sets. $$ The direction...
  7. T

    Relation between components and path-components of ##X##

    Homework Statement Theorem: If ##X## is a topological space, each path component of ##X## lies in a component of ##X##. If ##X## is locally path connected, then the components and the path components of ##X## are the same. I need help locating errors in my proof. Please help. Homework...
  8. CaptainAmerica17

    B Spivak's Calculus as a Prerequisite for General Topology

    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...
  9. mr.tea

    Topology What is a Good Supplementary Book for Topology Beyond Theorem, Proof?

    I am taking a course in topology with Gamelin and Greene, Introduction to topology. I would like to have some supplement to extend and give more motivation and explanation. I am quite tired of the "theorem, proof, theorem, proof" pattern. Thank you!
  10. mr.tea

    I Defining Neighborhoods in Topology: Inclusion vs. Containment

    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...
  11. Oats

    Topology Willard's General Topology vs Dugundji's Topology

    Hello, I have read a fair chunk of Munkres' Topology book and took a short introductory course during undergraduate, but I would like to learn point-set topology a little better. I have quite a bit of mathematical maturity, so that isn't an issue for me. I had a larger list of potential books to...
  12. lep11

    General topology: Prove a Set is Open

    Homework Statement Let A:={x∈ℝ2 : 1<x2+y2<2}. Is A open, closed or neither? Prove.Homework Equations triangle inequality d(x,y)≤d(x,z)+d(z,y) The Attempt at a Solution First I draw a picture with Wolfram Alpha. My intuition is that the set is open. Let (a,b)∈A arbitrarily and...
  13. A

    A Prep for Hawking/Ellis: Point Set Topology Needed

    I'm trying to prepare to read The Large Scale Structure Of Space-time by Hawking and Ellis. I've been reading a General Topology textbook since the authors say "While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavored to write this book so...
  14. beep300

    I General topology: Countability and separation axioms

    I need some help understanding the countability and separation axioms in general topology, and how they give rise to first-countable and second-countable spaces, T1 spaces, Hausdorff spaces, etc. I more or less get the formal definition, but I can't quite grasp the intuition behind them. Any...
  15. T

    General topology of a two terminal electrical device

    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...
  16. M

    Proof check: continuous functions (General topology)

    Homework Statement Let ##A \subset X##; let ##f:A \mapsto Y## be continuous; let ##Y## be Hausdorff. Show that if ##f## may be extended to a continuous function ##g: \overline{A} \mapsto Y##, then ##g## is uniquely determined by ##f##. Homework Equations The Attempt at a Solution...
  17. Hercuflea

    What Should I Know Before Taking General Topology?

    Hey everybody, I just wanted to ask a general question about Topology. I am planning on taking a General Topology course in Spring 2013 and first of all I don't know what it is. I am finishing up Differential Equations 1 right now with an A. By the spring I will have taken linear algebra 2...
  18. A

    What is general topology good for?

    I know that my question is not very clear, so I'll try my best to clarify it. Firstly, by general topology I mean point-set topology, because that's the only form of topology that I've encountered so far. In point-set topology, they teach us a lot of new definitions like open sets (that are...
  19. Fantini

    MHB Should I study metric spaces topology before general topology?

    Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which...
  20. H

    Should I take general topology or complex analysis?

    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...
  21. G

    Proving Disjoint Closed Sets in Metric Space: A Compact Set

    Homework Statement Suppose A and B are disjoint closed sets in the metric space X and assume in addition that A is compact. Prove there exists ∆ > 0 such that for all a ∈ A, b ∈ B, d(a, b) ≥ ∆ 2. The attempt at a solution I really don't have an attempt at a solution because I am 100%...
  22. Z

    Does Cocountable Topology Affect Countable Local Bases?

    Homework Statement Let X be a set equipped with a topology tau1, and let tau2 be the cocountable topology in which a set V in X is an open set if V is empty or X - V has only finitely or countably many elements. Consider the topology tau consisting of all sets W in X such that for each point p...
  23. H

    Any good problem book on General Topology

    I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract...
  24. T

    Best elementary 'topological spaces' in other words general topology book?

    Which would fit this description and with answers?
  25. E

    Proof of A Dense in Rn Not Bounded in Math

    I have the following A\subset\mathbb{R}^{n} is dense then A isn't bounded. Is this true? I know that A is dense iff \bar{A}=\mathbb{R}^{n} and that A is bounded iff \exists \epsilon>0\mid B_{\epsilon}(0)\supset A. How to proof it? Or there is an counterexample?
  26. quantumdude

    Relative Boundaries in General Topology

    Hi, I was trying to help a student with an assignment in topology when I was stumped by a symbol that I had not seen before. Here's the problem. a.) Let (X,\square) be a topological space with A\subseteq X and U\subseteq A. Prove that Bd_A(U)\subseteq A\cap Bd_X(U). The first thing...
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