Cover of a metric space.
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Let $E$ be a separable and complete metric space. Let $epsilon > 0.$ I want to find a cover of $E$ consisting of balls $B(q_n, epsilon), n in mathbbN, q_n in E,$ such that every $e in E$ is contained only in finitely many balls $B(q_n,epsilon).$ Is this possible?
metric-spaces
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up vote
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Let $E$ be a separable and complete metric space. Let $epsilon > 0.$ I want to find a cover of $E$ consisting of balls $B(q_n, epsilon), n in mathbbN, q_n in E,$ such that every $e in E$ is contained only in finitely many balls $B(q_n,epsilon).$ Is this possible?
metric-spaces
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up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $E$ be a separable and complete metric space. Let $epsilon > 0.$ I want to find a cover of $E$ consisting of balls $B(q_n, epsilon), n in mathbbN, q_n in E,$ such that every $e in E$ is contained only in finitely many balls $B(q_n,epsilon).$ Is this possible?
metric-spaces
Let $E$ be a separable and complete metric space. Let $epsilon > 0.$ I want to find a cover of $E$ consisting of balls $B(q_n, epsilon), n in mathbbN, q_n in E,$ such that every $e in E$ is contained only in finitely many balls $B(q_n,epsilon).$ Is this possible?
metric-spaces
asked Aug 7 at 15:03
White
305
305
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1 Answer
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I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.
add a comment |Â
up vote
2
down vote
accepted
I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.
I suppose that there is a more elementary proof than this one, but here it goes: every metric space is paracompact (this was proved by A. H. Stone) and every paracompact space is metacompact.
answered Aug 7 at 15:11
José Carlos Santos
115k1698177
115k1698177
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