## Reduction from minimum dominating set to the set cover

To solve the min dominating set problem of a graph G, we can reduce it to a set cover problem.

For example to find the MDS of the graph G:

We can create an instance of the Set Cover problem by:

1. Introducing the universe U to be equal to all vertices of the graph: U={0, 1, 2, 3, 4}.

2. For every closed neighborhood of a vertex in G, we introduce a subset S:

s= {{0, 1}, {0, 1, 2}, {1, 2, 3, 4}, {2, 3, 4}, {2, 3, 4}}

A minimum set cover solution of the instance (s, U) is similar to a minimum dominating set solution in the graph G.

My first question is how we can prove that this reduction is correct, and will give the correct MDS solution for every graph. In our example, the reduction works well and give the correct result.

My second question is how this reduction is related to the reduction when we deal with NP-complete proof(for example we can prove that the MDS problem is NP-complete by reduction to the vertex cover problem).

Are those reductions related? Thank you

## Dimension of a topological space equals the supreme of the dimension of its open cover

For a topological space $$X$$ which is covered by a family of open subsets $$\{U_i\}$$, then show that $$dim(X)=sup (dim(U_i))$$.

I understand that $$dim(X)\geq sup(dim(U_i))$$, so it only suffices to show that $$dim(X)\leq sup(dim(U_i))$$. Any help is appreciated!

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## Weighted Set Cover Problem Minimizing Average Weight

In the traditional weighted set cover problem, we aim at minimizing the sum of the weight of the selected sets. Is there any problem/literature that aim at minimizing the average weight of the selected sets (i.e., sum of the weight / number of sets selected)?

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## Can I use a hand holding a focus from one class to cover somatic components for a spell from another class?

I want to try playing a multiclass Hexblade 2 / College of Swords Bard 18, but I’m concerned about how focused will work.

I intend to hold a shield in one hand with a sword in the other. Thanks to College of Swords I am able to use the sword as a focus for my bard spells. However, I was hoping that I would be able to cast Shield from my hexblade spell list in the thick of battle.

As Shield requires a Somatic component, would I be able to use my sword hand (which is holding a focus, but for bard spells not warlock spells) to cover the Somatic components of Shield? I know that if it had a Material component I would need an arcane focus, but this situation seems ambiguous to me.

tl;dr Can I use my hand holding a bardic focus to cover the Somatic components for a Warlock spell?

pg. 15 XGtE under Bonus Proficiencies:

If you’re proficient with a simple or martial melee weapon, you can use it as a spellcasting focus for your bard spells.

pg. 203 PHB under Material (M):

A spellcaster must have a hand free to access these components, but it can be the same hand that he or she uses to perform somatic components.

This doesn’t actually state that materials are required for you to use the a hand holding the material components to perform somatic components.

I was really just hoping to get a response that cited a rule I might have missed (preferably about multiclassing, but such rules in the PHB skip over foci entirely).

## Book recommendation on Lie group which deals universal cover of Lie group

I am looking for a book about Lie group in a differential geometry point-of-view. Especially, I want a book that deals about the covering space of Lie group.

Actually, I think I have seen this kind of book before, but I don’t remember the title. However, I remember that Chapter 15 of this book has title “Lie groups and Lie algebras”(not certain), and there is a subsection(paragraph?) about the “Closed Subgroup Theorem”. Could you identify what the title of this book is?