What is General topology: Definition and 25 Discussions
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.
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...
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...
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...
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, +...
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...
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...
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...
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!
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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%...
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...
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...
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?
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...