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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2answers
55 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
2
votes
2answers
82 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...
0
votes
1answer
100 views

Homology of 3-sphere minus an embedding of $S^1 \times \mathbb{D}^2$

I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. ...
0
votes
1answer
43 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
2
votes
1answer
66 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
1
vote
0answers
121 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
0
votes
0answers
50 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
8
votes
2answers
131 views

A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
2
votes
2answers
103 views

A question on the proof of 14 distinct sets can be formed by complementation and closure

In Munkres, problem 20 of Section 2-6, it says that 14 distinct sets can be formed by complementation and closure. I see only five so far. Let f be the function of closure mapping and g be the ...
1
vote
1answer
40 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
3
votes
1answer
82 views

How to show a set is compact in a function space?

I have a question asking if $\{f_n\}$ is a compact in $C_b([0,\infty))$ (bounded continuous) with $||\cdot||_{L^\infty}$. The sequence is $$f_n (t) = \sin\sqrt{t+(2n\pi)^2},$$ I have showed that ...
1
vote
0answers
24 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
0
votes
1answer
166 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
3
votes
4answers
262 views

A set of all rational numbers in $[0, 1]$?

I have a question that is giving me some minor grief: If $A$ is a closed set containing all rational numbers $r \in [0, 1]$, then show that $[0, 1] \subset A$. I don't really understand this ...
-2
votes
1answer
93 views

Prove that intersection of connected spaces is connceted.

Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected. Using the definition of connected space is that the ...
5
votes
1answer
67 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
votes
0answers
32 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
vote
0answers
35 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 ...
1
vote
2answers
109 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 ...
2
votes
2answers
194 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 ...
2
votes
0answers
79 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 ...
1
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0answers
28 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
votes
1answer
48 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 ...
0
votes
1answer
39 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, ...
3
votes
2answers
248 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
71 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
45 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 ...
1
vote
1answer
57 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 ...
1
vote
1answer
48 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
67 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 ...
0
votes
1answer
82 views

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

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
vote
2answers
52 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
75 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
27 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
83 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
112 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
156 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
199 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
71 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
224 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
46 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
85 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
76 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
22 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
220 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 ...
2
votes
3answers
80 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 ...
13
votes
4answers
233 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
67 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
64 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
53 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 $i=2$ ...