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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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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238 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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466 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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58 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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62 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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212 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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127 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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337 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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107 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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92 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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29 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
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18 views

Intuitive affirmation on convex sets

Let $D_1, D_2$ two open, bounded and convex domain in $R^n$. Suppose that $D_2 \supset \overline{D_1}$, and the boundaries of these sets are of class $C^1$. Fix $x \in \partial D_1$ and suppose that ...
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16 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
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31 views

A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces

Prove that a local homeomorphism between compact, connected, Hausdorff spaces is a covering map of finite degree. Attempt at solution: Let $f:M\rightarrow N$ be the local homeomorphism. Since $N$ is ...
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23 views

Homotopic maps of a compact polyhedron

My friend and I are trying to solve the following exercise. Problem: Let $X \subset \mathbb{R}^n$ be a compact polyhedron. Show that there exists $\alpha > 0$ such that for any pair of maps $f, g ...
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30 views

Can we call the boundary of a subset of a topological space “partial X”?

Intuitively, one might be tempted to say $\partial S$ (the boundary of $S\subseteq X$ for X a topological space) as "partial X". Is this formally valid?
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21 views

Paracompactness and partitions of unity

For Hausdorff spaces, paracompactness is equivalent to finding subordinate partitions of unity for any open cover. I am confused about the "easy" step. If $f_j$, $j \in J$ is a partition of unity ...
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24 views

Show that $\cup A_n$ is connected.

Can someone please verify my proof or offer suggestions for improvement? Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \varnothing$ for all $n$. Show ...
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24 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
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23 views

A local homeomorphism between compact, connected, topological spaces

Prove that every local homeomorphism between compact, connected, topological spaces is a covering map of some finite degree. If the spaces were Hausdorff, the proof is easy, since then the singleton ...
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16 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
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29 views

In which of the three topologies does the following sequence converge?

Can someone please verify my proof or offer suggestions for improvement? Notations: $d(x, y) = |x-y|$: Standard metric on $\mathbb{R}$ $\bar d(x, y) = \operatorname{min}\{1, d(x, y)\}$: Standard ...
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33 views

On Compact Open Topology

Consider $X$ as a compact topological space and $Y$ as a metric space. Consider $C(X,Y)$, the set of all continuous functions from $X$ to $Y$. Prove that $C(X,Y)$ with compact open topology is induced ...
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28 views

Proof: Product Topology Question XxY

If $f$ is maps from topogical spacce $Z$ to $X\times Y$ so: $f$ is continuous iff : $\begin{cases} (p_X)\circ f: Z \rightarrow X\times Y \rightarrow X \\ (p_Y)\circ f: Z \rightarrow X\times Y ...
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52 views

A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
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43 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
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28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
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38 views

Proof about spectrum

Let X be a finite partially ordered set. How can to prove that there exists a ring R such that Spec R ≅ X? If anyone has any good way of thinking about them do please divulge..
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33 views

What does “single set” mean in this context?

I encountered this problem in Munkres topology. Let $X_1 , X_2$ denote a single set in topologies $\tau_1$ and $\tau_2$, respectively; let $Y_1 , Y_2$ denote a single set in the topologies $U_1, ...
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30 views

Urysohn's lemma and inf of rationals

In the course of the proof of Urysohn's lemma, one defines the function on the space X as the inf of a set of rational numbers that index sets containing each point x in X (except for f(F2) which is ...
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47 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...