# Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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### Show that every nearly compact space is almost compact space but the converse is not true

I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of ...
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### What are some interesting counterexamples given by finite topological spaces?

According to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book '...
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### Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
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### Does equal cardinality and one set being a subset of the other prove equality? [closed]

I'm currently solving a quite specific problem and in the final step I made a statement that can be generalised such that: $$((|A|=|B|)\wedge(A\subset B)) \implies (A=B)$$ Whilst this is clearly ...
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### Which groups $G$ has the property that for all subgroups $H$ , there is a surjective map from $G$ to $H$?

I tried many examples , but i can't find any counterexample . But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal ...
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### Example of a non semisimple Lie Algebra over $\mathbb{C}$ such that $L' = L$ [duplicate]

I wanted to show that if $L$ is a finite dimension semisimple Lie Algebra over $\mathbb{C}$, then $L' = L$ ($L' = [L,L]$) but the converse is not true. I could prove the statement but am unable ...
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### Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
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### Give an example for if … [duplicate]

Prove that if $H$ is a normal subgroup of a group $G$ and has index $n$, then for any $g\in G$ we have $g^n\in H$. Give an example for if $H$ is not normal, the mentioned statement is not correct. (...
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### substitutional interpretation of quantifiers: examples?

About the differences between propositional logic and (first order) predicate logic, given that if my basis is propositional logic I have to substitute the universal and existential quantifiers with ...
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### Propositional logic vs predicate logic: examples?

About the difference between the propositional logic and the (first order) predicate logic-> can you give me one or more remarkable examples which underly the differences and the similarities ...
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### Are there powerful ways to use the topological definition of continuity in real analysis?

In the lectures for introductory real analysis, my professor repeatedly told the class that the topological definition of continuity (preimage of open is open) is the most powerful version of ...
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### Group Operations/ Group Actions

I'm currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I'm reading the Artin textbook and I am not very clear of what exactly a group ...
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### Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive well-...
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### Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta$ set. What is it? Though there are ...
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### Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
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### Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
### Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded
Prove or Disprove, If $T: \mathbb R^n \rightarrow \ell_2$ is linear, then It preserves total boundedness $T(A)$ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators \$a_1, .., ...