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### If a graph has a perfect matching, does that mean that there is $S$ where $|S|$ is equal to $o(G-S)$?

In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings. Let ${\displaystyle o(G-S)}$ be the number ...
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### Definition of the structure sheaf on $\text{Spec} A$

In his book Algebraic Geometry, Hartshorne defines the structure sheaf of $\text{Spec} A$ to be the set of functions $s:U\to\coprod_{p\in U}A_p$ such that $s(p)\in A_p$ and $s$ is locally a quotient ...
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### Is Separable space iff compact space?

I'm reading chapter2 of PMA. In exercise 22~26, it uses separable space. According to exercises, compact$\implies$every infinite subset has a limit point$\implies$separable$\implies$countable base. ...
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### Cosets of alternating group in symmetric group

Let $G=S_n$ and $N=A_n$. There are exactly two left cosets of $A_n$ in $S_n$: $1\cdot A_n$ and $\left(1,2\right) \cdot A_n$, the latter consisting precisely of all odd permutations. Thus, the quotient ...
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### Are there any subsets of $\ \mathbb{R}\$ that have been investigated, but where it was not determined whether the set is countable or uncountable?
Are there any subsets of $\ \mathbb{R}\$ that have been investigated, but where it was not determined whether the set is countable or uncountable? In particular, I am thinking of a subset of \$\ \...