# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### Is the empty set linearly independent or linearly dependent?

Is empty set linearly independent or dependent?
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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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. $... 0answers 19 views ### General definition of the discriminant There is a notion of the discriminant of polynomials quadratic forms finite separable extensions of Dedekind domains (e.g., algebraic number fields) I don't know much about 2, but I think that 1 ... 0answers 53 views ### Why is the area of a rectangle given as$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$?" We define area as the "size of a two ... 1answer 31 views ### Is$\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{ ($x \in U$... 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 ... 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 ... 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$...
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 ...