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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210 views

Identity component of a Lie group

Could any one help me to solve this problem? Let the identity component $G_0$ of a Lie Group $G$ be the connected component of the identity element $e\in G$. Let $\mu$ and $i$ be the multiplication ...
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71 views

Uniform covers and partitions of unity

A cover $\mathcal C$ of a uniform space $(X,\mathcal U)$ is called a uniform cover if there is $U\in \mathcal U$ such that the cover $\{U(x):x\in X\}$ refines $\mathcal C$. Is it true that to every ...
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68 views

Adjoint to the Hom functor in Boolean rigs

What I wanna ask is about analogies to the tensor product for commutative boolean rings. What I mean by commutative boolean ring is set with two operations, + and *, and two identities, 0 and 1, as is ...
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128 views

Notation in Gilbarg/Trudinger? [Section 2.8]

Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
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154 views

topological group

Recently I'm interested in this open question: Must every star compact topological group be countably compact? star compactness ( which implies pseudocompactness ) = for every open cover $U$ of ...
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220 views

Properties of Topological Groups

I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
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124 views

Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
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87 views

Tangent bundle topology

Suppose $M \subset \mathrm{R}^n$ is a smooth manifold. Does the typical definition of the topology of the tangent bundle $TM$ of $M$ by charts coincide with the topology of $TM$ regarded as a subspace ...
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132 views

Definition of topologically onto map and Covering Spaces

In the chapter on Covering Spaces in his book A Basic Course in Algebraic Topology Massey uses the term topologically onto when defining covering spaces (see below for Massey's definition). What does ...
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161 views

Applications of monads in general topology?

What are applications of monads in general topology? For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
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87 views

Question related to some class of nowhere dense sets

Let $\Omega\subset (0,1)$ be a nowhere dense set which has no lower and upper bound in $(0,1)$ and for which $\Omega^{d}\cap(0,1)=\Omega$ ($\Omega^{d}$ denotes here the set of all limit points of ...
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53 views

Is this approach correct to show C*(X) is a topological ring?

Given that $C(X)$ (endowed with the m-topology whose base is given by $\left\{\{f \in C(X) : |g – f|\leq u\right\}: g \in C(X) \mbox{ and $u$ is a positive unit of }C(X)\}$) is a topological ring. Can ...
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161 views

Quotient map from $\mathbb R^2$ onto $\mathbb R^2\setminus \{(0,0)\}$

Could you help me to find a quotient mapping from $\mathbb R^2$ onto $\mathbb R^2\setminus \{(0,0)\}$? Assume the standard topology on both spaces. Thank you.
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95 views

Spotting the difference and improving rigor

What is the difference between these 2 questions? I have been asked to prove the following 2 cases -- (1), (2):  2 maps $f_1,f_2$, where $f_1:X\to Y, f_2:Y\to X$ and $f_1f_2$ is the identity map,  ...
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237 views

locally injective and continuous surjective function it´s homeomorphism

Let $f:M\to N$ be continuous and locally injective. If M is connected and exist a continuous function $g:N\to M$ such that $ fg= id_N$ then f is a homeomorphism from M to N. First clearly f is ...
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457 views

Open ball is an open set.

Could someone please show that an open ball is open where the definition of "open" is: A set is open if for each $x$ in $U$ there is an open rectangle $A$ such that $x$ in $A$ is contained in $U$. ...
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72 views

Preimages of 0 in antipode preserving maps on $S^n$

Attempting to find an inductive argument for the Borsuk-Ulam theorem led me to another question, which I found interesting in its own right but am stuck on. Let $g:S^n\rightarrow \mathbb{R}$ be a ...
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104 views

Do modules have any topology?

Is there any kind of topology, natural or unnatural, that modules do have? Is there any geometric interpretation for flat modules? Is "exactness" of a sequence, any kind of geometric condition? ...
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57 views

Every paraconvex set is convex if..

Observing that if we have a three-dimensional convex set then by sectioning it with a plane we obtain a convex set, I wondered if the converse is true: given a set whose every section is a convex set, ...
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58 views

Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
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156 views

Are topological vector spaces completely regular?

Every uniformizable space is a completely regular topological space. topological vector spaces are uniform spaces. every Hausdorff topological vector space is completely regular. From ...
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129 views

Equi-convergence of a function family at a point?

The definition of uniform integrability of a family of $L^1$ functions is: If μ is a finite measure, a subset $K \subset L^1(\mu) $ is said to be uniformly integrable if $\lim_{c \to \infty} ...
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46 views

Distributing pellets in multi-dimensional space

I'm designing a game with a multi-dimensional playing board, each dimension has the same length (an n-cube). Each of the dimensions wraps (think Asteroids). My goal is to place the pellets so they are ...
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101 views

Disjoint subsets in a compactification.

Let $X$ be the plane with the usual Euclidean metric, and let $cX$ be the compactification corresponding to the algebra of all bounded, uniformly continuous functions on $X$. I'm having trouble ...
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69 views

Making sense of values that decrese less and less as distance increases

I'm trying to come up with some formula to translate signal loss to distance. Distance from Transmitter = x Signal loss = y x=0 y=0 x=10 y=11.6 x=20 y=20.2 x=30 y=25.9 As you can see there is ...
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32 views

Stone spaces equivalent to projective limit definition

Is there a way to show that stone spaces are of the projective limit form $\varprojlim \mathcal{F}$ for a partially ordered set S and a map from $S^{op}$ to finite sets?
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76 views

x-section of closure of E of first category implies x-section of E nowhere dense

Let $E$ be a subset of first category of product space $X \times Y$. Why is the following true: if $(\bar E)_x \subset Y$ is of first category then it follows that $E_x$ is nowhere dense. $E_x$ ...
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192 views

How to understand $\limsup$/$\liminf$ of a subset of a complete lattice with a topology

From Wikipedia: Let $Y$ be a partially ordered set which is also a topological space and a complete lattice so that the suprema and infima always exist. For a set $X ⊆ Y$, define $$ ...
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211 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
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69 views

Projection maps of products of functors

Let I be a be a partially ordered set such that for any $i$; $i'$ $\in$ I, there exists $i''$ $\in$ $I$ such that $i'' > i, i'$. Let F be a functor from $I^{op}$ to finite ...
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170 views

homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $$ P_R^2 ...
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125 views

Continuity of a mapping of functions

Problem: A function $f : \mathbb{R} \to \mathbb{R}$ is said to be uniformly continuous if for all $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x_1)−f(x_2)| < \varepsilon$ ...
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67 views

connected sum of $n$ copies of $\mathbb{RP}^2$

$n\mathbb{RP}^2$ is $2n$-gon with identified edges. Can you check directions of arrows in picture? Thanks.
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34 views

Software for knotted $\mathbb{S}^2$'s in $\mathbb{S}^4$

According to the work of J. Scott Carter you can draw pictures of knotted surfaces in four-space in several different ways. I know the man is a real artist in this, but did anybody come across some ...
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335 views

Properties of the universal cover of CW-complexes

Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact: $X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
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93 views

Finite Levenshtein distance?

Is there a standard term for the relation on sequences where two sequences are related iff they have a finite Levenshtein distance, or for the equivalence classes it induces?
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165 views

how to factor a map by a group action

Let $X$ and $Y$ be topological spaces and a surjective map $f:X\rightarrow Y $. Suppose that a group $G$ acts on $X$. and let $\pi:X\rightarrow X/G$ be the quotient map. 1) Under what conditions $f$ ...
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133 views

One special set on [0,1]

Let $X = [0,1]$. Define $f:X\to\mathbb{R}_{\geq 0}$ to be Lipschitz continuous on $X$. Put $$Y\subset X:\int\limits_Y f(x)\,dx = 0$$ What can we say then about $A = X\setminus Y$? It is not defined ...
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106 views

reference search

Hello everyone I am looking for a couple of references: Claim 1 states that an open and connected set in $R^n$ is path-connected. Or more general an open, connected and locally connected set is ...
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91 views

Minimum rectilinear net

I'm looking for an algorithm to solve this problem: Given set of n points on 2D euclidean space, create a net of rectilinear edges, so that: 1. Every two points are connected with shortest edge. 2. ...
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27 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
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10 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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19 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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88 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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33 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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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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18 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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59 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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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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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, ...