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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4answers
36 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 ...
0
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0answers
13 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
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1answer
29 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
68 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 ...
1
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1answer
25 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: ...
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1answer
26 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", ...
-2
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0answers
21 views

Is subsapce topology from topology on Real projeective Space connected? [on hold]

$X = \{ [(0,x_1,x_2)] \in RP^2, x_1,x_2 \ne 0\}$ where $RP^2$ is real projective space. $\mathcal{T}$ is the subspace topology on X coming from the usual topology on $RP^2$ $X$ is connected, ...
0
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1answer
23 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
120 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
107 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 ...
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0answers
59 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
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1answer
20 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 $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
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2answers
57 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
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1answer
33 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
28 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
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1answer
24 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
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1answer
28 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 ...
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0answers
24 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
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0answers
18 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 ...
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1answer
47 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
36 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}) ...
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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
42 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
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0answers
23 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
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0answers
34 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
110 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 ...
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2answers
58 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
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1answer
43 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
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1answer
49 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 ...
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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
37 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 ...
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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
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1answer
23 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
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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 ...
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0answers
35 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
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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
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1answer
47 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 $D = \{ (x,x) \in I \times I \}$, ...
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3answers
84 views

Is the following set open?

$$S = \{ (x,y) \in \mathbb R^2\mid x^2 - y^2 < 1 \}$$ According to my geometry if we define $r$ as follows it should work, however I am having hard ideas proving it. Consider $z \in S$ and ...
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1answer
28 views

Accumulation points in C

Consider the following set $S = \{\frac{1}{n} + \frac{i}{m}: m,n \in \mathbb{N} \}$ I already got the accumulation points and proved that they are accumulation points of S $S` = \{\frac{1}{n}: n \in ...
3
votes
1answer
26 views

The set is closed (resp. open) iff the complement set is open (resp. closed)

There's a theorem in my small danish course book. Let $(M,d)$ be a metric space. Theorem: The concepts of open and closed are dual: A set $A\subseteq M$ is closed (resp. open) if and only if the ...
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2answers
22 views

Why are the sets $U_x$ disjoint in this proof of the non path-connectedness of the ordered square, $I_0^2$?

Let $p = 0 \times 0$ and $q = 1 \times 1$. Suppose there is a path $f: [a,b] \to I_o^2$ joining p and q. By the intermediate value theorem, $f([a,b])$ must contain every point $x \times y$ of ...
6
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1answer
57 views

Question 7.7 in measure theory on Radon measure from Folland's Real Analysis Second Edition

Hello all I was presented with this question from Folland's real analysis second edition on Radon measures which I am stuck on and so would really appreciate the help on. I m a novice in Radon ...
4
votes
1answer
62 views

Why is the Gromov-Hausdorff distance a metric?

The Gromov-Hausdorff distance is: $$ d_{GH}(A,B) = \inf_{f,g}d_H(A',B') $$where $f$ and $g$ are isometric embeddings of $A,B$ into some metric space, and their images are $A', B'$. The inf is taken ...
0
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2answers
35 views

Complex Analysis ( Open/Closed Set).

let $z = re^{i\theta}$ , How do we prove that , $0\leq \operatorname{arg}(z)\leq\dfrac{\pi}{4}$ ($z \neq 0$), is neither a open set nor a closed set. $\operatorname{arg}(z)$ is nothing but $\theta$ ...
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0answers
72 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
1
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0answers
46 views

Every open set is the union of net of increasing open sets

I'm struggling to find a solution to the following problem: Let $(X,\mathcal{T})$ be an arbitrary topological space and let $\mathcal{U}$ be an class of subsets of $X$, i.e. ...
1
vote
0answers
30 views

Help with proving the following: Open set $U$ in normed space $X$ is connected iff it is connected by polygonal lines.

Help with proving the following: Open set $U$ in normed space $X$ is connected iff it is connected by polygonal lines: $\forall a,b \in U , \exists P_n \subseteq U , P_n(a,b) $ I would like if ...
2
votes
1answer
63 views

Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function such that for every $x \in \mathbb{R}$ there exists $n$ such that $f^{(n)}(x) = 0$, then f is a polynomial. I'm kind of lost on this ...
0
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0answers
32 views

Separated subsets of $\mathbb{R}^k$

Let $A$ and $B$ be separated subsets of some $\mathbb{R}^n$, suppose $a\in A, b\in B,$ and define $p(t)=(1-t)a+tb$ for $t\in \mathbb{R}^1$. Put $A_0=p^{-1}(A), B_0=p^{-1}(B)$. (a) Prove that $A_0$ ...