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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3
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1answer
43 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
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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
30 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
67 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
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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
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1answer
26 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
52 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
87 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 ...
0
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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
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1answer
27 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
24 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 ...
7
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1answer
60 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
63 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
78 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
47 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. ...
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
33 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 $A_0$ ...
0
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2answers
72 views

Can someone please explain to me why the set {1, 1/2, 1, 1/3, 1, 1/4, …} has two points of accumulation?

A point of accumulation in $X$ is a point $c$ where every neighborhood of $c$ contains at least one point of $X$ distinct from $c$ For example, the neighborhood of $1$ is $\{1, 1/2\}, \{1,1/2,1\}, ...
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1answer
46 views

Prove that $\{(x,y)\mid x\in\mathbb Q\}\cup\{(x,y)\mid y\in\mathbb Q\}$ is a connected subset of $\mathbb R ^2.$ [duplicate]

In my notebook, something is very briefly, not in detail whatsoever, path connectedness mentioned, and two assumptions are made about $x_1, x_2 \in \mathbb Q.$ If anyone can prove this I would greatly ...
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3answers
44 views

Constructing true metrics in infinite dimensional vector spaces?

Is there an example of a true metric defined on a function space? I'd imagine it is some type of integral involving two functions, and it will return a value that obeys the metric axioms, but I have ...
0
votes
4answers
74 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
3
votes
2answers
80 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
0
votes
2answers
46 views

Analyzing this topological space.

Consider the following space: the underlying set is $C = C_1 \cup C_2$, where Ci is the circle of radius $i$ and centre $0$ in the complex plane. Basic open sets are: • $\lbrace z\rbrace$ for every ...
5
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1answer
83 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
0
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1answer
34 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
4
votes
1answer
47 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
4
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0answers
43 views

Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
0
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1answer
41 views

How to show $G$ is a perfect set that contains no rational points?

For $E:=[0,1]$, since $\Bbb Q\cap E$ is enumerable, let it be $\{q_1,q_2,\cdots\}$. If I remove the elements of $V_1:=(q_1-\frac1{10},q_1+\frac1{10})$ from $E$, I obtain a closed (and compact) set ...
1
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2answers
147 views

Lang's treatment of product of Radon measures

Let $X$ be a locally compact Hausdorff space. We denote by $\mathcal B(X)$ the $\sigma$-algebra of Borel sets of $X$. A positive Radon measure $\mu$ on $X$ is a measure defined on $\mathcal B(X)$ with ...
0
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1answer
56 views

Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..

using anyone of the following definitions(and no other concerning compactness): -$A \subseteq (X, \tau)$ is compact if for every open cover of A there exists a finite cover. -A compact set in a ...
1
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1answer
32 views

Need help with this question concerning compact spaces

Let the set be given like in the following manner: $$\{x_n: n\in\mathbb N\}\subset \mathbb{R^n}$$ $$l^2=\left\{\{x_{n}\}_{n=1}^{\infty}\,\Big|\, \sum_{n=1}^{\infty}|x_n|^2<\infty\right\}.$$ Prove ...
2
votes
2answers
55 views

If $a \in \mathbb{I}$ , how is $\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$

If $a \in \mathbb{I}$ , how is $$\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$$ It says in my notebook that this set in dense in $\mathbb{R}.$ How do I prove this density? With say $\mathbb{Q}$ and ...
2
votes
1answer
74 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
0
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2answers
75 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
4
votes
2answers
56 views

Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with discrete topology has uncountably many points?

How do I show that the Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with the discrete topology has uncountably many points? There is a hint that crux is to construct a ...
1
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1answer
68 views

Is simply connectedness is a topological property?

A topological space $X$ is called simply-connected if it is path-connected and any continuous map $f:S^{1} \to X$ (where $S^1$ denotes the unit circle in Euclidean 2-space) can be contracted to a ...
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1answer
68 views

An example for uncountable compact space

Would someone please give an example of a space which is compact but not countably compact space? Is my example right? : suppose there exist a collection of sets ${\{S_i}\}$ for all $i\in \mathbb ...
1
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1answer
57 views

Locally connected Locally compact separable metric space

Let $X$ be a locally connected locally compact separable metric space. Is it possible to find a countable collection $\mathcal{B}$ such that every member of $\mathcal{B}$ is a nonempty peano subspace ...
0
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0answers
35 views

n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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0answers
25 views

Peano space - A compact connected locally connected metric space

Let $X$ be a peano space. Is it possible to find a countable collection $\mathcal{\mathbb{B}}$ which forms a base for $X$ and every member of $\mathcal{\mathbb{B}}$ is a peano subspace of $X$.
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1answer
48 views

With what additional assumption, would a connected space be path connected?

Let $X$ be a connected space. What additional condition on $X$ would imply that $X$ is path-connected? The only one I know is by assuming $X$ is an open subspace of a normed space $V$. What else ...
4
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3answers
50 views

Topology: Prove that this subspace topology is discrete

The question is from Topology and Its Applications Chapter 1, by William F. Basner. The question states the following, Let $\mathbb{Z}$ be a topological space with subspace topology inherited from ...
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1answer
73 views

Suppose every convergent Sequence has a unique limit point in space $X$, then $X$ is Hausdorff

My attempt Let $a, b \in X$ are two distinct points . we will show that there exist two open sets $G_1$ and $G_2$ such that $a \in G_1$ and $b \in G_2$ and $G_1 \cap G_2 = \phi$ or there exists $n ...
1
vote
2answers
64 views

Accumulation points in $R^2$ and general $R^n$

So I am solving at the moment question of this form first if we take this question. Determine the accumulation point in $R^2$ and if they are closed sets ,open sets, or neither. All complex numbers z ...
0
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0answers
13 views

Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
0
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1answer
50 views

Proof that the Order topology on $\mathbb{R}$ has the same basis as the Euclidean topology

I want to prove that the Order topology on $\mathbb{R}$ has the same basis as as the Euclidean topology on $\mathbb{R}$. Assume that the only thing we know about the order topology is that it has the ...
2
votes
1answer
57 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
0
votes
1answer
36 views

Show that $A\subseteq B\implies A^{\circ} \subseteq B^{\circ}$ in a different way.

Let $A$ and $B$ be subsets of a metric space $(M,d)$. If $A\subseteq B$, then $A^{\circ} \subseteq B^{\circ}$. Proof : Assume that $a\in A^{\circ}$. Then there exists a $r>0$ such that ...