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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5
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1answer
37 views

Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that ...
0
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0answers
16 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 ...
1
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0answers
17 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
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2answers
79 views

A proof that if the product of spaces is Hausdorff, each of them is Hausdorff

Is my approach to this question right? Question: Prove that if $$\prod_{\alpha \in J} X_\alpha (\neq \emptyset) $$ is Hausdorff, each $X_\alpha$ is Hausdorff. Attempt to answer: It is enough to ...
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2answers
32 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
1
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0answers
42 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
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0answers
15 views

Mapping open on open dense subset => Mapping is open on whole space?

Let $X,Y$ be topological spaces, and let $f\colon X \to Y$ be a continuous function. Further suppose that there exist an open and dense subset $U$ of $X$, such that $f\vert_{U} \colon U \to Y$ is an ...
0
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1answer
40 views

Does the boundaries of non-disjoint sets in Euclidean space have common element?

I've got stuck while solving a problem, and the thing I need is; If you are given two open sets in $\mathbb{R}^{n}$, where they have both common element and non-common element. (That means, their ...
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1answer
31 views

If $A=[0, 1] \times (0, 1)$, which is a subspace of $I^2 = [0, 1] \times [0, 1],$ how are the sets $U_x = \{x\} \times (0, 1)$ open in $A$?

If each of $U_x$ is open, doesn't this imply that $\{x\}$ is open in $[0, 1]$, which contradicts the uncountability of $[0, 1]$? This question arises from an example (#5) in James Munkres' Topology, ...
2
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2answers
46 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
0
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0answers
53 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 ...
2
votes
1answer
30 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
0
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1answer
32 views

Open Finite Cylinder homeomorphic to $\mathbb{R}$?

That was an exam question asking for the homeomorphism between: $\mathbb{S}^1 \times (a,b)$ and $\mathbb{R}$. My guess: since $(a,b)$ is homeomorphic to $\mathbb{R}$, function $\mathbb{S}^1 \times ...
0
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1answer
25 views

Difference between the cone and open cone

What is the difference between the cone $$CX=X\times [0,1]/X\times \{0\}$$ and the open cone $$OC(X)=X\times [0,1)/X\times \{0\}?$$ I mean what is done by taking $[0,1)$ instead of $[0,1]$.
1
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1answer
37 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
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1answer
57 views

Is there an uncountable scattered subset of $\mathbb R$? [on hold]

Does anyone know if there exists an uncountable scattered subset of $\mathbb R$? A set $A$ is scattered iff it contains no non-empty subset which is dense-in-itself (i.e. every subset $F$ of $A$ ...
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2answers
50 views

Can a countably infinite set be dense in $\mathbb{R}^2$? [on hold]

Can a countably infinite set be dense in $\mathbb{R}^2$? In other words, what is the topological density of $\mathbb{R}^2$?
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2answers
35 views

A trouble with the discrete product topology

Consider a finite set $S=\{1,\ldots,n\}$ with the discrete topology, and moreover construct the product topological space $S^\mathbb N$ with the product topology. $S^\mathbb N$ is made by all the ...
0
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0answers
67 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
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1answer
23 views

Question regarding a wording of an exercise related to Noetherian topological space

The exercise states "If $X$ is a Noetherian topological space, show that the union of any subset of the connected components of $X$ is always open and closed in $X$." Does the question mean "If I ...
0
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2answers
68 views

${\overline{A}}^{\circ}= \varnothing \Longrightarrow {\overline{A \cap B}}^{\circ}= \varnothing $

If $X$ is a topological space and $A,B,C \subseteq X$ with $B \subseteq A$, I am wondering if the following statements are true. $(1)$ ${\overline{A}}^{\circ}= \varnothing \Longrightarrow ...
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3answers
48 views

The union of finitely many closed sets is closed

I understand how the definitions unpack for this proof, but I'm not sure how to formally word it. Let $(X, \tau$) be a topological space. We say that $A \subseteq X$ is closed if $X\setminus A \in ...
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2answers
108 views

An equivalent characterisation of open subset of a topological space

I'm having trouble understanding topologies. We say that $U \subseteq X$ is open if $U \in \tau$. If $(X, \tau)$ is a topological space and $U \subseteq X$, why are these properties the same? $U$ ...
5
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1answer
57 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
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3answers
38 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
10
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1answer
150 views

On different definitions of neighbourhood.

I am going through the basics of topology, mainly to refresh them. I had taken a course some years ago but never used topology actively. So I am reading Munkres's Topology. I have noticed that he ...
2
votes
1answer
42 views

Proof of a distance

I have one distance shown as an example in a book but I'm striving to demonstrate that it is effectively a distance. here it is said : let $U=\{z\in\mathbb{C, |z|=1}\}$ we can get a distance on $U$ ...
4
votes
1answer
39 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
0
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1answer
27 views

Open and closed equivalence relations

I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function ...
0
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0answers
11 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, ...
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2answers
139 views

Existence of Open Covers

Do sets always have open covers exist? I know they are not always finite, but do infinite ones always exist? I was reading baby rudin and the proofs for non-relative nature for compactness seems to ...
1
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3answers
48 views

Show that $Int(A)=X\setminus\overline{X\setminus A}$.

Let $A$ be subset in topological space of $(X,\tau)$, show that $\operatorname{Int}(A)=X\setminus\overline{X\setminus A}$. The definition of interior provided is "largest open set in A", which I ...
11
votes
3answers
147 views

Fundamental group of the product of 3-tori minus the diagonal

I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 ...
2
votes
2answers
54 views

Embedding of $\mathbb{R}^2 \to \mathbb{R}^3$ with non-parallel tangent planes

I have a qual question here and I'm struggling to get a good starting point. The question asks to construct a smooth proper embedding $f\colon \mathbb{R}^2 \to \mathbb{R}^3$ such that for any distinct ...
2
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1answer
51 views

Is it true that $A$ is scattered?

Let $X$ be a (Hausdorff) topological space and for each ordinal $\alpha$ denote by $X^{(\alpha)}$ the $\alpha$th derivative of $X$ by the Cantor-Bendixson derivation (i.e., define transfinitely: ...
2
votes
2answers
47 views

Is the product of $T_i$ spaces always a $T_i$ space?

I am doing some topology and wondering about the following. If $X_j$ is a $T_i$ space for some $i \in \{1,2,3,3.5,4\}$. Does it then follow that $\Pi_j X_j$ is again a $T_i$ space? I think for $i=2$ ...
1
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1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
9
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5answers
327 views

Why are box topology and product topology different on infinite products of topological spaces?

Why are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...
1
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1answer
43 views

Show that $\overline{A\cap B}\subseteq\bar{A}\cap\bar{B}$

Let $A,B$ be subset of a topological space, show that $\overline{A\cap B}\subseteq\bar{A}\cap\bar{B}$. (The bar denotes closure) I have totally no clue, please give me some idea.
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1answer
67 views

d' is finer than d on compact space $\implies$ $\forall \epsilon \ \ \exists \epsilon' \ \ \forall x \ B_{d'}(x,\epsilon') \subseteq B_d(x,\epsilon)$

This is my conjecture, but I guess I am missing the key idea for the proof (or my conjecture is wrong) Let d and d' be two metrics on a compact space $X$ ($X$ is compact with respect to both ...
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63 views

why some solid can have no surface? [on hold]

For solid construction, I can understand the closed surface has no edges. But i cannot understand why some solid can have no surface (except just lines?), any other solid which can have no surface?
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1answer
35 views

Fixed point of continuous mapping between punctured disk

Let $X=B^2-\{a\}-\{b\}$, where $B^2$ is the unit disk on $\mathbb{R}^2$, $a, b$ are interior points of $B^2$. Is there a continuous map $f:X\rightarrow X$ which has no fixed point? Thank you.
4
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1answer
72 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
4
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1answer
49 views

Product of T1 spaces is T1

I am trying to prove that the product of T1 spaces is also T1. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} = ( ...
3
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0answers
65 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
2
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2answers
49 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
2
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1answer
28 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
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5answers
386 views

homeomorphism non-example

A homeomorphism is a continuous function between topological spaces that has a continuous inverse function. Can someone provide examples of a continuous function between topological spaces that does ...
1
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1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
0
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0answers
46 views

Homework on open covering and partitions of unity [closed]

a) If $\{U_1,U_2,\ldots,U_n\}$ is a finite open covering of the normal space $X$,prove that there exists a partition of unity dominated by $\{U_i\}_i$.