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Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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Question about quotient of a compact Hausdorff space

I am reading the book 'Algebraic Topology' by Tammo Tom Dieck. On page 12 in the proposition 1.4.4 he states that : Let $X$ be a compact Hausdorff space and $f : X \rightarrow Y$ be a ...
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I have these two definitions for an adherent value of a sequence the first is : $a$ is a an adherent value for $(x_n)$ iff $$\displaystyle \forall \varepsilon>0,\forall n\in \mathbb{N},\exists ... 1answer 69 views The set A := \{ (x,y) : y \in f(x,x+1) \} is closed if f : \mathbb{R} \to \mathbb{R} is continuous. Let f : \mathbb{R}\to \mathbb{R} continuous. I need to prove that A := \{ (x,y) : y \in f(x,x+1) \} is closed in \mathbb{R^2}. We have that \mathbb{A} is the union of all the ... 1answer 151 views A problem in A Course in Point Set Topology by Conway, union of totally bounded sets This is stated as a problem in A Course in Point Set Topology book by J. Conway: Let \{E_n\} be a sequence of totally bounded sets. If \operatorname{diam}E_n\to 0 as n\to\infty, show that \... 1answer 68 views null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity I am reading a book suggesting that the following is true: Let X be a simply connected (maybe finite) simplicial complex. Then there exists a continous map$$g : X \to X,$$homotopic to the ... 1answer 69 views Show that the set is a closure of U_j Let \Omega be an open and nonempty set and \Omega \subset \mathbb{R}^n. Let's define a set U_j=\{x\in\Omega:\Vert x\Vert<j \wedge dist(x,bd(\Omega))<\frac{1}{j}\}. We observe that U_j ... 1answer 165 views Jingle River (Berbed wire) Metric Problem I want to prove that Jungle River metric is indeed a metric space, and determine it is open and closed balls. Firstly, i know that the metric is given by x,y\in \mathbb{R}^2, such that x=(x_1,x_2), ... 2answers 298 views Closed or open + proof Proof if H = \{ (x,y)\mid 0<x<1, 0<y<1\} is closed, open or neither. So I would strongly suggest that it is open because if you draw it you recognize that the complementary set ... 2answers 55 views A question about two common definitions Two definitions make me puzzled ! 1. The definition of \textbf{Functions Differentiable at a Point}: A function f defined in a neighborhood (x_{0}-\delta,x_{0}+\delta)of a point x_{0}, ... 1answer 244 views Definition of continuity in topological spaces does not seem quite right. Here's how continuity is defined in most standard topology texts A function from X to Y is continuous iff the inverse image of each open set of Y is open in X. This definition does not ... 1answer 49 views Topologial properties of f([1,3]^3) where f(x,y,z)=x^2+2xz+y If f(x,y,z)=x^2+2xz+y, determine f([1,3]^3) and characterize this set in terms of openness, closedness, completeness, compactness and connectedness. Since [1,3]^3 is compact then f([1,3]^3) ... 1answer 41 views Decompostion into countably many nowhere dense compact sets Let A be a meagre subset of a locally compact abelian Polish group G. Then A can be written as a countable union of nowhere dense subsets of G. Is it always possible to write A as a ... 2answers 48 views Opens in a manifold I am studying for my exam of manifolds and I dno't understand why the following is true. Let (U,\phi) be a local map at p in the manifold M, and let \phi(p)=0. Then there exists an open set ... 2answers 796 views Proof that there is no continuous 1-1 map from the unit circle in \Bbb R^2 to \Bbb R. Let S^1=\{(x,y) \in \Bbb R^2 : x^2+y^2=1\}. I'm trying to prove that there is no 1-1 continuous mapping between S^1 and the real line. The map is not necessarily onto. Proof so far: Suppose such ... 2answers 62 views Open/Closed value of unions of Open/Close intervals. I hope I do not make a duplicate here but I couldn't find this question on here nor a clear answer on the internet. So let's say I have two intervals, A and B. Let's define C as the union of A... 1answer 108 views Definition of disconnected subsets in metric spaces and in more general settings I found the following paragraph in a Real Analysis book (namely, Carother's one). A subset E of a metric space M is disconnected in E if there exist disjoint, nonempty, open (in E) ... 2answers 68 views could a spanning tree graph be expressed by a lower triangular matrix? Suppose a directed spanning tree graph G, there are n nodes, and the root is node 1. We express this graph by a matrix M_{n\times n}. If there is an directed edge from node i to node j, ... 1answer 67 views Which is finer(larger) between the sequence spaces l_{p} & l_{p+1} Prove that, l_{3}\subset l_{7} & L_{9}[0,1]\subset L_{6}[0,1], where l_{p} & L_{p}[0,1] are of usual notation. Are the converses hold for both cases? Can these two results ... 1answer 73 views Level set of a continuous function strictly increasing in each argument Let F : \mathbb{R}^d \to [0,1] be absolutely continuous and strictly increasing in each argument. Is it true that the boundary of the set \{ \boldsymbol{x} \in \mathbb{R}^d: F(x) \geq \alpha \} ... 1answer 58 views Show there is a path in X \times Y if and only if there is a path in X and a path in Y (a) Let (x_1,x_2) \in X and (y_1,y_2) \in Y. Show that there is a path from (x_1,y_1) to (x_2, y_2) if and only if there is a path x_1 to x_2 in X and a path y_1 to y_2 in Y. (b) ... 1answer 48 views the openness of the union of two disconnected sets (one of which is open and the other one is closed) Now we have two disconnected sets in complex plane, one of which is open and the other one is closed. Is the union of these two sets open? 1answer 64 views Does B(H) satisfy in Heine-Borel property? Based on here, I know that every bounded and closed subset of a space is not compact. I really want to know that B(H), the space of bounded linear operators, satisfies in Heine - Borel property. ... 1answer 106 views Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable? Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ... 0answers 31 views Upper hemicontinuity of a correspondence I would like to know whether the following correspondence is upper hemicontinuous:$$ C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases}, $$... 1answer 61 views Why X contains a countable \pi-basis? I don't understand the following statement. First, I write what a \pi-bases means. Let X be a topological space and \mathcal{B} a family of non-empty open sets. We call \mathcal{B} a \pi-... 0answers 69 views What is meant by saying that two paths in X from x_0 to x_1 are equivalent? Suppose that X is a topological space and x_0, x_1 are points of X. What is meant by saying that two paths in X from x_0 to x_1 are equivalent? I presume it's enough to just say the path ... 1answer 185 views Motivating the compact-open topology It has been a while since I studied algebraic topology, and I wanted to revisit homotopy theory. Determined to take a more sustainable approach, I started by questioning and verifying every result in ... 2answers 202 views What is the one point compactification of the reals? In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by 0? I vaguely remember something about a ... 1answer 243 views Limit points and boundary sets in topology The main difference between an open set and a closed set is a closed set includes its boundary while an open set does not. However, in topology, a closed set is also distinguished (from an open set) ... 2answers 158 views Closure of an open set in manifold I have a question. During a proof of a proposition, the following is stated: Let K be a compact set in a manifold M of dimension n. Then there exists an open set U such that K\subset U and ... 1answer 11 views balls have empty boundary with regard to the p-adic norm Let p be prime, a\in\mathbb{Q} and r\geq0. How can I show that the closed ball D(a,r) in (\mathbb{Q},|\cdot|_p) must have an empty boundary (with regard to the topology induced by the p-... 0answers 50 views Space generated by a reflection Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ... 1answer 36 views does closed under sequence limits imply the set is closed? In a topological space T we have a set F such that the limit of every convergent sequence of elements of F is in F. can we deduce that F is closed? if T be second countable then it's true,... 2answers 50 views Topology on k((t)) k((t)):=\lbrace (a_i)_{i \in \mathbb{Z}}, a_i \in k,\exists \ N \in \mathbb{Z} \ s.t \ \forall \ i<N, a_i=0\rbrace where k is a field of char zero. We define componentwise addition and ... 2answers 484 views Let X and Y be topological spaces. A function f: X → Y is continuous if and only if f^{-1} (C) is closed in X for every closed set C ⊂ Y. I need help proving this theorem This is the first part of the biconditional, I think if I can prove this. Proving the converse shouldn't be nearly as difficult. Assume f is continuous ⇒ f^{-1} (C)... 2answers 72 views If f is a continuous function from R^3 to R and K⊂R^3 is compact, show that there exist two points a, b ∈ K so that f(K)⊂[f(a),f(b)] If f is a continuous function from R^3 to R and K⊂R^3 is compact, show that there exist two points a, b ∈ K so that f(K)⊂[f(a),f(b)]. When is f(K)=[f(a),f(b)]? What I believe is the ... 1answer 74 views Prove that Open Sets in \mathbb{R} are The Disjoint Union of Open Intervals Without the Axioms of Choice There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable? 2answers 67 views Manifold with \pi_1(M)=F_n We may construct a 3-manifold M_n with \pi_1(M_n)\cong F_n (i.e. the free group on n generators) as follows: consider the complement of n pairs of open 3-balls in \mathbb{R}^3. For each pair,... 1answer 62 views When does a topological group embed topologically in its group of homeomorphisms? Let X be a topological group. X acts freely on itself by left multiplication; this gives us an injective group homomorphism \Phi: X\rightarrow \operatorname{Homeo} X. Under what conditions is \... 1answer 61 views Every point of an open ball is a centre for the open ball. Suppose X is a nonempty set and d is an ultrametric on X i.e.,$$d(x,y)\le\max\{d(x,z),d(z,y)\}$$for all x,y \in X. Suppose B is an open ball of (X,d). Show that every point of B is a ... 0answers 59 views How to prove a function is harmonic polynomial 1! How to prove this function a harmonic polynomial using Laplace equation For the 1 question I know we can prove harmonic using Laplace Equation but for this on m confused how to start. For the 2 ... 2answers 754 views Is sum and product of a infinite number of continuous functions are also continuous functions? Whether in Real Analysis or by Open Set Def of Continuity in Topology, it is easy to show that the sum and product of a FINITE number of continuous functions are also continuous functions. That is, ... 0answers 51 views Is the question in the Munkres's topology book wrong? At the end of cheapter 8.1, 4) Given spaces X and Y, let [X,Y] denote the set of homotopy classes of maps of X into Y. b) Show that if Y is path connected, the set [I,Y] has a ... 1answer 67 views Dense set meaning on this proof… Theorem: Let f and g : X → Y be continuous functions (Open Set Definition of Continuity). Assume that Y is Hausdorff and that there exists a dense subset D of X such that f(x) = g(x) for ... 3answers 36 views S^{n+m+1} can be decomposed as the union of S^n\times D^{m+1} and D^{n+1}\times S^m along their boundaries. Let S^n denote the n-sphere and D^n denote the n-disk (of course, \partial D^{n+1}\cong S^n). Then S^n\times D^{m+1} and D^{n+1}\times S^m both have boundary S^n\times S^m. The ... 1answer 212 views Proving a nowhere vanishing vector field on 2D manifold implies TU\cong M\times S^1 So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle TU is ... 2answers 193 views Continuous function on complete bounded metric space need not be bounded I came across the following old qual problem: Suppose (X,d) is a complete metric space with finite diameter. Is every continuous function on X bounded? It seems like the function 1/x on ... 1answer 38 views Find the map of the closed ball B(0,1) of the following continuous function f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1) and f^{-1}(0). Find the map of the closed ball B(0,1) of the following continuous function$$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right) and $f^{-1}(0)$. $f^{-1}$ seems quite simple, I got $(0,2,-9)$,...
Show that the intersection of two connected sets is connected if the two sets are disjoint. Is the set $1\leq x^2+y^2+z^2 \leq 9$ connected and/or compact? I think its compact because it's closed ...