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; ...

learn more… | top users | synonyms (3)

0
votes
0answers
83 views

Disconnected Sets definition and connectedness of the unit interval

The definition of a disconnected set seems a bit ambiguous in the book I am reading : $1.$ A subset $D$ of $\mathbb R^p$ is said to be disconnected if there exist two open sets $A$ and $B$ such ...
0
votes
0answers
29 views

Each infinite subspace of a KC-space …

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
0
votes
0answers
33 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
0
votes
0answers
20 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
0
votes
0answers
121 views

A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces

Prove that a local homeomorphism between compact, connected, Hausdorff spaces is a covering map of finite degree. Attempt at solution: Let $f:M\rightarrow N$ be the local homeomorphism. Since $N$ is ...
0
votes
0answers
37 views

Homotopic maps of a compact polyhedron

My friend and I are trying to solve the following exercise. Problem: Let $X \subset \mathbb{R}^n$ be a compact polyhedron. Show that there exists $\alpha > 0$ such that for any pair of maps $f, g ...
0
votes
0answers
35 views

Can we call the boundary of a subset of a topological space “partial X”?

Intuitively, one might be tempted to say $\partial S$ (the boundary of $S\subseteq X$ for X a topological space) as "partial X". Is this formally valid?
0
votes
0answers
32 views

Paracompactness and partitions of unity

For Hausdorff spaces, paracompactness is equivalent to finding subordinate partitions of unity for any open cover. I am confused about the "easy" step. If $f_j$, $j \in J$ is a partition of unity ...
0
votes
0answers
36 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
0
votes
0answers
18 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
0
votes
0answers
54 views

On Compact Open Topology

Consider $X$ as a compact topological space and $Y$ as a metric space. Consider $C(X,Y)$, the set of all continuous functions from $X$ to $Y$. Prove that $C(X,Y)$ with compact open topology is induced ...
0
votes
0answers
54 views

Proof: Product Topology Question XxY

If $f$ is maps from topogical spacce $Z$ to $X\times Y$ so: $f$ is continuous iff : $\begin{cases} (p_X)\circ f: Z \rightarrow X\times Y \rightarrow X \\ (p_Y)\circ f: Z \rightarrow X\times Y ...
0
votes
0answers
69 views

A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
0
votes
0answers
47 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
0
votes
0answers
30 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
0
votes
0answers
42 views

Proof about spectrum

Let X be a finite partially ordered set. How can to prove that there exists a ring R such that Spec R ≅ X? If anyone has any good way of thinking about them do please divulge..
0
votes
0answers
51 views

What does “single set” mean in this context?

I encountered this problem in Munkres topology. Let $X_1 , X_2$ denote a single set in topologies $\tau_1$ and $\tau_2$, respectively; let $Y_1 , Y_2$ denote a single set in the topologies $U_1, ...
0
votes
0answers
35 views

Urysohn's lemma and inf of rationals

In the course of the proof of Urysohn's lemma, one defines the function on the space X as the inf of a set of rational numbers that index sets containing each point x in X (except for f(F2) which is ...
0
votes
0answers
33 views

Hausdorff topology of a set of subsets

In the text I'm reading, there is a map from the C, the complex plane to E, a collection of compact subsets in C. Continuity with respect to the Hausdorff topology on E was talked about and I'm ...
0
votes
0answers
68 views

Confused by definition of an open set in “All the Mathematics You Missed”

On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$: A set $U$ in $\mathbb{R}^n$ will be open if given any $a ...
0
votes
0answers
14 views

Presheaf of real valued functions

Seen as how a Presheaf of real valued functions on a topological space X associates a function f:U→ℝ to each open set U, what function maps the empty set to ℝ since the empty set is by definition an ...
0
votes
0answers
50 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
0
votes
0answers
63 views

Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
0
votes
0answers
50 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
0
votes
0answers
58 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
0
votes
0answers
97 views

Proof that a correspondence is upper hemicontinuous if and only if it's graph is closed

I'm working through a textbook (General Equilibrium Theory) where proofing the following theorem is left as an exercise to the student - unfortunately I dont know how. Theorem 23.1: (A ...
0
votes
0answers
22 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
0
votes
0answers
49 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
0
votes
0answers
32 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
0
votes
0answers
71 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
0
votes
0answers
75 views

Topology problem: Proving that sections are open

I have been trying to learn some basics of topology on my own, I have learnt the basic definitions. I have not been able to understand the proof provided in the text. Could anyone provide a clearer ...
0
votes
0answers
22 views

Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
0
votes
0answers
53 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
0
votes
0answers
122 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ ...
0
votes
0answers
51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
0
votes
0answers
39 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
0
votes
0answers
17 views

Infimum simple function and stepfunction

If $A_n$ is a simple function and $B_n$ is a Stepfunction, then the infimum ($A_n\wedge B_n$) is a stepfunction. Why is this true?
0
votes
0answers
38 views

$\omega_1$ contains an initial segment as a copy of $\omega + 1$

Here, $\omega + 1$ is defined to be the union of Natural number and a number larger than all natural numbers. I know that the number larger than all natural number in this set works as a limit point ...
0
votes
0answers
48 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
0
votes
0answers
18 views

What is the (pre)topology hinted by the Daniell integral

Daniell takes a vector lattice $H$ of a set $\mathbb{R}^X$ which he calls the set of elementary functions. For him, an elementary integral $I$ is a nonnegative functional on $H$ which verifies : if ...
0
votes
0answers
23 views

Name for point in an infinite product of intervals bounded away from the boundary

Consider $[0,1]^{\omega}$ (the product of a countable amount of $[0,1]$ intervals). I am interested in points $x=(x_1,x_2,\ldots)\in [0,1]^{\omega}$ that are bounded away from the boundary, i.e. ...
0
votes
0answers
37 views

Need help with the proof of the KKM-lemma.

I have been working on the proof of the KKM-lemma, which states Let $\lbrace A_0,A_1,...,A_n \rbrace$ be a closed covering of an $n$-simplex $\sigma=[x_0,...,x_n]$ such that for each face ...
0
votes
0answers
55 views

Order of refinement of an open covering of $X$, a metric space

If every finite open covering of a metric space $X$ has a refinement of order $\leqslant n$, is it true that every open covering does too? We say that a covering has order $n$ if $n$ is the largest ...
0
votes
0answers
30 views

The number of non-degenerate proper subcontinua in a non-degenerate continuum

A continuum is any compact connected metric space. A continuum is non-degenerate if it is not a single point. My question is thus this: How many non-degenerate proper subcontinua must a ...
0
votes
0answers
23 views

Is there a term in Topology for a boundary point that belongs to a given set?

If M is a subset of a topological space and U is the interior of M, is there a standard name for the set M - U? In other words, is there a standard name for the set of boundary points of a given set ...
0
votes
0answers
46 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
0
votes
0answers
27 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
0
votes
0answers
43 views

Relational calculus. Natural join

How does natural join look in relational calculus? I think it looks like {A1, A2, A3, A4 | r1(A1, A2, A3) AND r2(A2, A3, A4)} Correct me please if I am wrong. ...
0
votes
0answers
28 views

A question about the Zariski space.

A Zariski space is a topological space with the property that every descending chain $F_1\supset F_2\supset F_3\dots$ of closed sets is eventually constant. Show that every Zariski space can be ...
0
votes
0answers
56 views

finite subset topology

Let $X$ be a set and $\tau=\left\{X\right\}\,\cup\,\{u_i\subset X|u_i \text{ is finite}\}$. Is $\tau$ is a topology on $X$? My effort to show this is as follows: 1) $X\in\tau$ by definition and ...