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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2
votes
2answers
42 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
votes
0answers
46 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
27 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
votes
1answer
30 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
votes
1answer
23 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
vote
1answer
32 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 ...
-1
votes
1answer
56 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 $A$ ...
-1
votes
2answers
49 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 $\mathbb{R}^2$?
1
vote
2answers
34 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
votes
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
votes
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
votes
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 ...
0
votes
3answers
46 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 ...
1
vote
2answers
107 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
votes
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 ...
1
vote
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
votes
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
36 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
votes
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
votes
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, ...
1
vote
2answers
133 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
vote
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
139 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
53 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
votes
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
45 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
vote
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 ...
8
votes
5answers
312 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
vote
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.
2
votes
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 ...
-1
votes
0answers
62 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?
0
votes
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
votes
1answer
71 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
votes
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} = ( ...
2
votes
0answers
60 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
votes
2answers
48 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
votes
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 ...
7
votes
5answers
383 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
vote
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
votes
0answers
46 views

Homework on open covering and partitions of unity [on hold]

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 $\{U_i\}_i$.
2
votes
0answers
23 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
2
votes
1answer
60 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
0
votes
1answer
56 views

A detail in the proof of Jordan's theorem

The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere ...
5
votes
1answer
54 views

A maximal subset of $S^2$ with respect to a connectedness property

Let the set $A$ be a circle with a chord on the sphere $S^2$. Obviously $A$ has the following property: P: $\quad$ Any two points $a$ and $b$ of $A$ can be connected by a path that ...
0
votes
0answers
32 views

Disconnecting using totally disconnected sets [duplicate]

Let $X$ be $[0,1]^2$ and $S\subset X$ a totally disconnected subset. Is it true that $S^c$ is always connected? If it is false, what can we say when $X=[0,1]^n$?
2
votes
1answer
53 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
2
votes
2answers
90 views

Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous.

Let $d$ be a metric on $X$ and let $A$ be any arbitrary subset of $X$. Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous. Let $p\in X$. We want to show that for any ...
1
vote
2answers
45 views

Show that for any subsets $A,B\subset X$: (i):$d(A\bigcup B)\leq d(A)+d(B)+d(A,B)$ and (ii) $d(\bar A)=d(A)$

Let $d$ be a metric on $X$. Show that for any subsets $A,B\subset X$: (i) $d(A\cup B)\leq d(A)+d(B)+d(A,B)$ (ii) $d(\bar A)=d(A)$ I found this hard to prove because the diameter ...
-2
votes
0answers
33 views

Does an lp norm induce a ball topology? [closed]

Namely, does the metric $$||x - y||_p$$ induce the usual ball topology that a metric induces? I wasn't able to find any results regarding this on a quick Google search.