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

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Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?

Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all $U \in \mathcal{T}, x \in U \implies D \cap U \...
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1answer
34 views

What should I call an “injective” algebra?

Given rings $A,B$, we say that $B$ is an $A$-algebra if there is a ring homomorphism $f:A\rightarrow B$. This homomorphism give the structure of the algebra. Various properties of algebras can either ...
2
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1answer
59 views

Writing higher order derivatives using the limit definition of derivative?

Studying Taylor series, I wanted to get a sense for what higher derivatives really express in precise terms using the limit definition of the derivative. Is this correct? $$\frac{d^2y}{dx^2} = \...
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1answer
39 views

Complementarity and Substitutability

I am reading a paper in the international journal of game theory entitled Unequal Connections by Goyal and Joshi (2006) and it has the following sentences: "If strategic complementarity obtains... In ...
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2answers
40 views

Some doubts on the definition of minimality of a graph. EDITED

If a graph $G$ is minimal with property $P$, does that specifically mean: Any proper subgraph of $G$ does not have that property $P$? Or, Any graph with less number of vertices or less number of ...
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0answers
78 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
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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
27 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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21 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 ...
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3answers
192 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
481 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
38 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
604 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
24 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
47 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 $...
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3answers
104 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
189 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
56 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
30 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
26 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
28 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
98 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
2
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2answers
34 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
76 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)=...
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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
558 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
62 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: $$\...
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1answer
52 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 ...
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4answers
487 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
59 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
50 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 ...