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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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. ...
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18 views

Show this a Open Cover of (0,1)

I'm going through some material for this real analysis course (again) and stumbled upon this open covering of (0,1): $\kappa_n = (\frac{1}{n},1-\frac{1}{n})$. Since, for $n \to \infty$, $\frac{1}{n} ...
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4 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
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20 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 $\mathbb{R}^2$. ...
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2answers
177 views

Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell

Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent. I think it can be proved by showing they are both deformation retracts of ...
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18 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 $S^n$. ...
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1answer
185 views

Generalization of Banach's fixed point theorem

I wanted to show that if $f:X\to X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ ...
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1answer
19 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 ...
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2answers
45 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 ...
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1answer
42 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 ...
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96 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 ...
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79 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 ...
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41 views

Why Must the Degree of this Map be 0? [on hold]

Let $f:S^3 \times S^1\times S^1\rightarrow S^3 \times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$.
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1answer
428 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
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45 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 ...
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48 views
+50

Creating a Topological group from modulo multiplication Group.

If I were to create a Topology out of the Modulo 3 Multiplication group $\mathbb{Z}_3$, what elements would it consist of and why? So $\mathbb{Z}_3 = \{0,1,2\}$ as a group over modulo 3. What are the ...
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376 views

Homotopy Question Help?

Let $X$ be a topological space and suppose $X_1$ and $X_2$ are spaces obtained by attaching an n-cell to $X$ via homotopic attaching maps. Show that $X_1$ and $X_2$ are homotopy equivalent. Proof: ...
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355 views

Two spaces homotopy equivalent to eachother, attaching maps, Algebraic Topology.

I have a question regarding algebraic topology with which I was hoping someone could help me with. I've managed to show the following: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ ...
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1answer
58 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 ...
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1answer
28 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 ...
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1answer
25 views

If $S$ is dense in $L^{2}$. Is it true that $pS=\{pf| f\in S, pf\in C^{\infty}\} $ is dense?

Let $S=\{f\}$ be a set of function defined in a compact subset $\Omega\subset \mathbb{R}^{n}$ such that $S$ is dense in $L^{2}(\Omega)$. Is it true that for $p\neq 0$ a rational function $pS=\{pf| ...
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1answer
18 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 $X$ ...
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17 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 ...
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1answer
58 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 ...
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2answers
49 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 ...
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4answers
87 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 ...
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2answers
181 views

Is a connected, first countable space necessarily Hausdorff?

Is a connected, first countable space necessarily Hausdorff? I've been trying for forever to come up with a counterexample but haven't had any luck.
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2answers
85 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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1answer
17 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
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1answer
40 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 ...
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2answers
102 views

Arcwise connected but not connected?

In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces: With a few pathological exceptions, arcwise connectedness is practically ...
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23 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 ...
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0answers
18 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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1answer
44 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
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48 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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17 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 ...
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1answer
41 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, ...
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1answer
213 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
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2answers
142 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 ...
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2answers
772 views

compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
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4answers
895 views

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components ...
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1answer
39 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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2answers
47 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 : ...
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1answer
31 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 ...
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57 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 ...
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3answers
148 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 ...
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1answer
35 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 ...
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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 ...
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1answer
26 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 $[0,1]$.