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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6
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1answer
104 views

What star domain has a non-star-domain interior?

Definition: We call a subset $S$ of $\mathbb{R}^n$ a star domain (or star-shaped) if there exists a point $x_0 \in S$ such that for every $x \in S$, the line segment $\overline{x_0x}$ is contained ...
0
votes
1answer
24 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
1
vote
1answer
24 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
0
votes
0answers
21 views

Plotting Distance Constrained Points on a Plane

Does anybody know of some algorithmic way to tell if it is possible to plot a set of distance constrained points on a cartesian plane. Or, better still, a method to determine the minimum number of ...
1
vote
1answer
28 views

If $A\subseteq\Bbb R$ is nonempty with $|A|\ge 2$, then $A$ totally disconnected $\iff A^\circ=\emptyset$.

In the course of working on an exercise, I came up with the claim given in the title. Just looking for verification. $\underline{\text{Claim: } A\text{ is totally disconnected}\iff ...
1
vote
2answers
49 views

Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
0
votes
1answer
32 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
0
votes
2answers
20 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
1
vote
2answers
39 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
1
vote
0answers
47 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
2
votes
2answers
48 views

If $A$ is path connected, then $\bar A$ is path connected?

I know the topologist's sine curve serves as a counter example. But how do I show that $A = \{(x, \sin (1/x)): 0<x\le 1\}$ is path connected?
5
votes
1answer
50 views

Approximating nice functions with wild ones

Let $X$ and $Y$ be toplogical spaces, and call a function $f:X\to Y$ wild if the preimage $f^{-1}(\{y\})$ is dense in $X$ for every $y\in Y$ -- or, equivalently, if the image of every nonempty open ...
8
votes
2answers
80 views

Is the following a characterization of $\Bbb Q\cap\cal C$, where $\cal C$ is the Cantor set?

Let $A$ be an ordered set, with the following properties: $A$ is countable $A$ has a least and greatest element Between any two points with successors are points without successors; between any two ...
1
vote
4answers
79 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
1
vote
1answer
31 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
0
votes
1answer
34 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
6
votes
1answer
75 views

Is there an infinite topological meadow with non-trivial topology?

For reference meadows are a generalization of fields that were designed to be compatible with the requirements of universal algebra. Specifically a meadow is a commutative ring equiped with an ...
2
votes
1answer
29 views

A covering space of CW complex has an induced CW complex structure.

Let $X$ be a $CW$ complex, and let $q : E \rightarrow X$ be a covering map. Prove that $E$ has a $CW$ decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$. Hint: ...
0
votes
1answer
28 views

Product-topology of discrete $\{ 0, 1 \}$ spaces

I was thinking about the following: Take the product $\prod_{i \in I} \{ 0, 1 \}$ for each $\{ 0, 1 \}$ being discrete. Is the product-topology also the discrete topology? I'd say intuitively "no", ...
0
votes
1answer
24 views

Please check if my reasoning about whether this topological space is connected is correct

$X = \mathbb{R}$, $\mathcal{T} =$ the collection of all subset $U$ of $\mathbb{R}$ such that $U = \emptyset$ or $\mathbb{R} - U$ is finite. Then, $(X,\mathcal{T})$ is connected. My thoughts: assume ...
1
vote
1answer
121 views

Cauchy Sequences--is the floor function of a Cauchy sequence also a Cauchy sequence?

Okay so say you have some Cauchy sequence (a_n). And c_n=[[a_n]], where [[x]] refers to the greatest integer less than or equal to x. Is c_n also a Cauchy sequence? This is what I've got so far, ...
7
votes
2answers
112 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
-2
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0answers
60 views

How to prove that multiplication is continuos function? [on hold]

How to prove that multiplication is continuous function? $x \rightarrow x^n$ Can somebody help me? :)
4
votes
1answer
28 views

$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
1
vote
2answers
59 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
3
votes
1answer
36 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
0
votes
1answer
30 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...
0
votes
1answer
25 views

Is a characteristic map in CW complex a quetient map?

Let $X$ be a CW complex and $\Phi : D \rightarrow \bar e$ be the characteristic map for an open cell $e$. I wonder whether $\Phi$ is a quotient map. I konw it is surjective. But I cannot prove that ...
1
vote
1answer
30 views

Prob. 2, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: Compactness of $[0,1]$ in the lower limit topology

Let $\mathbb{R}_l$ denote the set of real numbers with the topology having as a basis all the half open intervals $[a,b)$ on the real line. Then is the closed interval $[0,1]$ compact as a subspace ...
0
votes
0answers
25 views

Prob. 1, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: An infinite subset of $[0,1]^\omega$ without limit points in the uniform topology?

Let $[0,1]^\omega$ denote the set of all sequences of real numbers in the closed unit interval $[0,1]$, and let the uniform metric $d$ on $[0,1]^\omega$ be given by $$d\left( (x_n)_{n\in\mathbb{N}} , ...
1
vote
0answers
20 views

Specific problem on Radon measures from Folland's real analysis on $ C_0(X) $

Hello all I am trying to understand the concept of $ C_0(X) $ within the concept of Radon measures as presented in Folland's real analysis chapter 7, so far so good right until I came across problem ...
1
vote
1answer
48 views

Confused about the open/closed set in metric space

Let $(M,d)$ be a metric space. I understand well that $\emptyset$ and $\mathbb{R}$ are both open and closed sets. I read some notes that say, that $\emptyset$ and $M$ are both open and closed. So, ...
0
votes
1answer
37 views

Connected Components are either equal to each other or have nothing in common. Any Hint is appreciated

The connected component definifiton $(X, \mathcal{T}_X)$ topological space. $(A, \mathcal{T}_A)$ subspace topology. Let $x\in X$ $ C_x:= \bigcup_{, A\subset X, x \in A, (A, \mathcal{T_A}) ...
0
votes
1answer
20 views

Connected components and showing subsets are equal

If $Z_1, Z_2 \subset X $ are connected components, show that $Z_1 = Z_2$ or $Z_1 \cap Z_2 = \emptyset$ Note: we defined connectedness as a splitting of two open sets $U_1$, $U_2$ such that$U_1 = X$ ...
3
votes
1answer
43 views

Distance between a point and an empty set: meaning and value?

On page 253 in General Topology by R Engelking: The distance $\rho(x, A)$ from a point $x$ to a set $A$ in a metric space $(X,\rho)$ is defined by letting $\rho(x, A) = \text {inf}\ {\{\rho(x, a) ...
2
votes
0answers
25 views

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where…

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where $$\mathbb{I_r}=[x_0-r,x_0+r]$$ and $$\mathbb{P}=\{(x,y): |y-y_0|\leq a, |x-x_0|\leq b\}\subset \mathbb G $$ where $\mathbb G-$ ...
5
votes
0answers
37 views

Question on Radon measure's Lebesgue decomposition

Hi all seeing as how people were so nice to me and my experience was a success I though perhaps it was safe to try and ask this as well on Radon measures (also same class) I am given a $ ...
2
votes
2answers
112 views

Is $\bigcup_{n=1}^{\infty}\left ( -1+\frac{1}{n},1-\frac{1}{n} \right )$ open?

This .pdf on Example 2 (page 4 on paper), it says that $$\bigcup_{n=1}^{\infty}\left ( -1+\frac{1}{n},1-\frac{1}{n} \right )=\{0\}\cup (-1/2,1/2)\cup\dots=(-1,1)$$ is open. Please check Theorem 1 on ...
1
vote
2answers
60 views

What is the homeomorphism between a disk and an ellipse?

A disk/circle is defined by $$C = \{(x,y) \in \mathbb{R^2} : x^2 + y^2 \leq r^2\}$$ An ellipse is defined by $$E = \{(x,y) \in \mathbb{R^2}: x^2/a^2 + y^2/b^2 \leq 1 \}$$ How can we define a ...
4
votes
1answer
44 views

Question on Radon measures from Folland's Real Analysis

Greetings my mathematical friends. I am taking a summer class on measures and the theory of real analysis, and I was given the following question from Folland's Real Analysis Second Edition Chapter 7 ...
0
votes
1answer
52 views

Prove that $(a,b]\subseteq \mathbb{R}$ is not open.

I want to prove myself that a half-interval $(a,b]\subseteq \mathbb{R}$ is not an open set. I checked it in here. My proof: We wish to prove that $b\notin (a,b]^{\circ}$. Assume that $b\in ...
0
votes
0answers
48 views

Confusion regarding continuous functions between topological spaces – a subtle but possibly important point

Let $T: V_1 \to V_2$ be a linear mapping. Show that $T$ is a continuous function between $(V_1, \tau_{V_1}) $ and $(V_2, \tau_{V_2}) $ A direct solution to the problem is not what I am looking ...
2
votes
1answer
39 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
0
votes
1answer
38 views

Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
3
votes
1answer
24 views

Coincidence of two $\tau$-additive measures

I'm struggling to prove the following Lemma from V.I. Bogachev, Measure Theory 2: Let two $\tau$-additive measures $\mu$ and $\nu$ on a topological space $X$ coincide on all sets from some class ...
7
votes
2answers
65 views

Density of a dense subspace of a Hausdorff space

If X is a Hausdorff space and Y is a dense subspace of X, can the density of Y exceed the density of X? The density of a space X is the least infinite cardinal C such that X has a dense set of ...
0
votes
0answers
36 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
2
votes
1answer
25 views

The restriction fo covering to a component is a covering map onto its image.

I am reading Lee's Introduction to Topological Manifolds. I got stuck on the problem 11-7 on pages 303. The below is the problem. Prove : If $q: E \rightarrow X$ is a covering map and $A \subseteq ...
3
votes
1answer
32 views

Convergence that preserves smoothness

One of the advantages of uniform convergence is that it preserves continuity (among other properties). Unfortunately, it does not preserve derivability. Is there a convergence mode preserving it?
4
votes
1answer
49 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...