# Tagged Questions

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

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### Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
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### What are pullbacks of finite-coproduct injections along arbitrary morphisms?

I am studying a definition of an extensive category: An extensive category is a category $E$ with finite coproducts such that pullbacks of finite-coproduct injections along arbitrary morphisms exist ...
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### What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to formulate the definition but succeeded only ...
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### What is the “Donkey Theorem”?

I was watching the Turkish version of Who Wants to Be a Millionaire? and they asked this question: What field is the Donkey Case (or I guess it can be translated as Donkey Theorem) related to? ...
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### How is the set of all even numbers definable from $\omega$?

This Set Theory textbook (page 89) defines definable sets as follows: Definition 6.8. Given a set $a$ and a formula $\Phi$ we define the formula $\Phi^a$ to be the formula derived from $\Phi$ by ...
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### Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"
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### Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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### Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
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### Definition of extrema in calculus of variations

I am reading Gelfand and Fomin about Calculus of Variations and in page 12 they say: ' Analogously, we say that the functional $J[y]$ has a (relative) extremum for $y=\hat{y}$ if $J[y]-J[\hat{y}]$ ...
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### What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing ...
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### What exactly are the meaning of the followings in the definition of a category?

In Awodey's Category Theory a category is defined as follows. A category consists of the following data, Objects: $A, B, C,\ldots$ Arrows: $f,g,h,\ldots$ For each arrow $f$ there are ...
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### What is necessity for integral to be well-defined in defining solutions?

When trying to give a notion of solutions for differential equations with non-local terms, e.g., integral of unknown functions, to guarantee that the integral is well-defined, i.e., finitely ...
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### Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
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### What does it mean to say “a divides b”

I am not a number theorist and I am learning about relations. I encountered a relation that says $a \leq b$ if $a$ divides $b$ Can someone clarify what it means to a number to divide another ...
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### Open vs Closed Immersions of Locally Ringed Spaces

I'm reading Qing Liu's book at the moment and I'm trying to figure out why open immersions of locally ringed spaces are required to be isomorphisms on stalks, but closed immersions are only required ...
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### What is the difference between an adherent point and a limit point?

What is the difference between an adherent point and a limit point? Please explain this with a proper example.
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### Difference between NP-hard and NP-complete

I am struggling to tell the difference between the definitions of NP-hard and NP-complete problems. I know that NP-complete problems are NP-hard, so this tells me that \text{$P_1$ polynomially ...
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### How would you define a 3d angle?

Today I somehow was wondering how could you define a 3d angle. First what I could think of was volume of a cone with side a=1. Then if it wasn't round, volume of a tetrahedron with side 1. Is any of ...
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### What is the difference between accumulation point and $\omega$ accumulation point?

The title says it all. Accumulation point has a widely known definition: a point in $X$ is accumulation point if every open set containing $x$ contains infinitely many points of $X$ Sometimes I ...
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### Why can projection function on $X \times S$ be regarded as a local homeomorphism?

I am studying some properties of local homeomorphism I am in particular trying to find a local homeomorphism that is not a homeomorphism and the projection function seems to be the perfect candidate ...