For requesting, clarifying, and comparing definitions of mathematical terms.

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16 views

Sheaf definition vs “Mayer Vietoris”

Let $F$ be a presheaf on a space $X$ and say that $F$ has property MY if for all $U, V$ open in $X$ we have an exact sequence $$0 \to F(U \cup V) \to F(U) \oplus F(V) \to F(U \cap V)  $$ Is this ...
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0answers
28 views

What are the components and quasicomponents?

I'm trying to understand the difference between components and quasicomponents. I'm using the following definitions: $x\sim y$ iff $x$ and $y$ lie together in some connected set. The component of ...
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0answers
22 views

Are these correct extension of injective and surjective functions?

Recall the definition of injective and surjective functions: 1.a A function $f: X \to Y$, is injective if for all $a,b \in X$, if $a \neq b$, then $f(a) \neq f(b)$ 1.b A function $f: X \to Y$,...
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2answers
32 views

if $g$ is a lie algebra what is $g^*$?

Iam trying to learn what a coadjoint orbit is but I can't since everywhere I look the definition involves $g^*$.Something that I googled and didn't find anything. I am not even sure what $g^*$. Is it ...
2
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3answers
194 views

How many “co-” topologies are out there?

So far I have learned about $\tau_{co-finite}$ and $\tau_{co-countable}$ Are there any other co-related topologies like...$\tau_{co-infinite}$? In general, what is the condition we need to have a co-...
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4answers
490 views

Is the empty set linearly independent or linearly dependent?

Is empty set linearly independent or dependent?
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2answers
31 views

Is Reliability Component a vertex?

The term component has a distinct definition in graph theory from vertex while the terms components and vertices can be mostly the same in Realiability Engineering, my intuition. So how is the term ...
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2answers
90 views

Are integers relevant for every Group?

The definition of the order of an element in a group is: The order of an element $x$ of a group $G$ is the smallest positive integer $n$ such that $x^{n}=e$. Doesn't this definition assume that ...
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2answers
41 views

What does the condition of $T_{3\frac{1}{2}}$ space mean exactly?

A topological space $(X,\mathcal{T})$ is said to be $T_{3\frac{1}{2}}$ if given $x \in X$, and a closed set $C \subset X$, $x \not \in C, \exists f:X \to [0,1]$ s.t. $f(x) = 0, f(C) = 1$ This ...
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5answers
610 views

Question about definition: what is an affine linear space?

In an article I am reading it says : Let $H$ be an affine linear space of codimension $m$... Could someone please explain me what is meant by affine linear space? Thanks!
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25 views

Tangent space in the book “Differential Forms and Applications”

In the book "Differential Forms and Applications", the author defines the tangent space of $\mathbb{R}^{3}$ at $p$ ($p \in \mathbb{R}^{3}$) as $\mathbb{R}^{3}_{p}=\{q-p; q \in \mathbb{R}^{3}\}$. My ...
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1answer
26 views

Proving that a sequence is Cauchy 5

We want to prove that the sequence $a_n = n^2$ is Cauchy in the metric space $(E, d)$, with $E = [0, \infty[$ and $d(x, y) = |\arctan(x) - \arctan(y)|$. I proceed in the following way: $a_n$ is ...
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2answers
89 views

Is $|x| = -x$ true for $x = 0$?

What are the solutions for this equation? $|x| = -x$ It is clear for me that all negative numbers will fulfill this (my brother doesn't believe me, but that doesn't matter). But I'm having a ...
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1answer
26 views

Lerch Transcendent Notation and Definition

I used Wolfram-Alpha to try to find the sum $$\sum_{n=0}^\infty (-1)^n\ln\frac{n+2}{n+1}.$$ While it only gave a numerical result, it showed the partial sum formula included $\text{LerchPhi}^{(0,1,0)}...
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1answer
25 views

Definition of $n$-to-$1$ mapping.

What is the definition of "$n$-to-$1$ mapping"? Does an $n$-to-$1$ mapping mean to say that if $f$ is a function from $A$ to $B$, then for every $y\in R(f)$ there exists $n$ different elements in $A$ ...
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0answers
48 views

Locality and base change along effective descent morphisms

Definition. Let $\mathcal M$ be a class of arrows in a site $\mathsf{C}$. An arrow $\pi:E\rightarrow B$ is said to be locally in $\mathcal M$ if there's a covering ${u_i:U_i\rightarrow B}$ such that $...
6
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3answers
107 views

Quotient of a graph?

I want to understand quotient of a graph (also called quotient graph), my teacher says that the terms quotient of a graph and a modulo of a graph should be synonyms (even though modulo of a graph ...
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0answers
39 views

Alternative view of matrix inversion (explanation required)

We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we ...
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1answer
41 views

Different definitions of a limit of a function in $\mathbb R$. Are they equivalent or not?

If a limit $L$ of a function $f:A\to\mathbb R$ exists at a point $a\in \mathbb R$, where $A\subset\mathbb R$ is a proper subset of the set of real numbers, is there any difference between the ...
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2answers
190 views

Is the reverse statement to “open” in Munkres topology true?

There is a problem: Given $(X, \tau), A \subseteq X, \forall x \in A, \exists U \in \tau, x \in U \text{ s.t. } U \subset A \implies A$ is open in $X$ So what I did was to show that $A$ is in $\...
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0answers
58 views

Definition of regular map from aquasi projective variety

These are from the book Basic Algebraic geometry by Shafarevich Definition of regular function on a quasi projective variety is as follows : Let $X\subset \mathbb{P}^n $ be a quasi projective ...
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1answer
31 views

What are associated and non-associated polynomials?

Studying algebraic geometry (if it makes any difference) I've found the following "... here the $F_i$ are irreducible and non-associated polynomials... ". It must be obvious what it means for ...
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2answers
16 views

Is this triangulation consistent with Sperner's Lemma?

Since any two triangles which intersect have an edge or a vertex in common, the triangulation is simplicial. However, I am concerned about triangle $A$. Is every sub-triangle supposed to have a ...
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1answer
29 views

What is the definition of subcollection in topology?

I had thought that subcollection referred to a collection of sets in some topological space $(X,\tau)$, where each set is a subset of $X$ But I also see it being used in subbasis, and subspace ...
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1answer
16 views

Definition of bounded in a metric space - confirmation

Is the following definition of a bounded metric space correct? $(M,d)$ is bounded if $\exists a \in M, r > 0$ such that $M = B(a,r)$. Looking around on the internet I instead see $M \subset B(...
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0answers
29 views

Clarification of Sperner's Lemma

From Graph Theory by Bondy, Murty Image from wikipedia I don't see how the picture holds according to the definition from the Graph Theory book. Specifically, the definition says to assign ...
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3answers
113 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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2answers
36 views

Does “sphere” denote the surface or the entirety of a solid ball?

In everyday English, the word "sphere" denotes a 3-dimensional object, including the points inside the surface and its center. However, I get the sense that in mathematics, the sphere is used ...
4
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1answer
79 views

What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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3answers
105 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle f(x)=...
4
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3answers
53 views

Vectors sometimes used in math just as arrays/lists of numbers, sometimes as concept of “change”

As a freshman in a small town college. Ive been getting mixed signals to what vectors (and matrices/tensors) are. Sometimes I get the feeling they are used just as containers/arrays for multiple ...
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4answers
563 views

Why do they call it base 10?

Now, I know intuitively why it's called base 10: because there's 10 numbers. But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at 9....
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0answers
20 views

What is characteristic time?

What is characteristic time? Where is it useful? From this answer by Joriki: The characteristic time is usually defined to be the time in which a quantity decreases by $1/e$. Why is ...
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0answers
22 views

Continuously variable *space* [closed]

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
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1answer
48 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) -...
3
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1answer
34 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a "...
2
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2answers
63 views

Is continuity at a point only defined for points in the domain?

I'm using Michael Spivak's Calculus, 3rd edition textbook. Without ado, I'll state the definition given for continuity at a point: DEFINITION$\;\;\;\;$The function $f$ is continuous at $a$ if: $$\...
2
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1answer
53 views

Is a hypersurface really defined by an arbitrary polynomial?

In An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...
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0answers
28 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
5
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4answers
493 views

Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
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0answers
44 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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3answers
65 views

changing the order of the logical symbols $(\forall \epsilon) (\exists\delta)$ by $(\exists \delta)(\forall\epsilon) $ in limit definition

Some time ago a professor told the class, which I was in, to analyze why this definition of limit is not good (or if it is a good definition to argument why): There exists a $\delta>0$ for all $...
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1answer
51 views

What is the meaning of (resp. closed) in set theory?

I'm sure this a spectacularly basic question but I can't seem to find the definition of this anywhere. Here's some context: If $U$ and $V$ are open (resp. closed) then $U\cup V$ is open (resp. $...
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19 views

General definition of the discriminant

There is a notion of the discriminant of polynomials quadratic forms finite separable extensions of Dedekind domains (e.g., algebraic number fields) I don't know much about 2, but I think that 1 ...
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53 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
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1answer
31 views

Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ ...
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1answer
19 views

Definition of vertex-cut for digraph?

I am trying to understand vertex cut for digraph. I could find this for graphs Vertex cut is a vertex whose removal increases the number of components in a graph. (D67, Handbook of Graph Theory by ...
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1answer
32 views

Definition of component for a digraph?

I could find this in Wikipedia Component: A connected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have ...
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0answers
49 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
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25 views

Showing that an intersection of indexed sets is a subset of every individual indexed set

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$. This seems obvious to me from ...