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Show that in a compact metric space there are at most a countable number of clopen sets
Non-separable compact spaceProve that a separable metric space is Lindelöf without proving it is second-countableEquivalent Metric Using Clopen SetsOpen balls with radis $>epsilon$ in a compact metric spaceSubset of separable metric space can have at most a countable amount of isolated pointsExample of a locally compact metric space which is $sigma$-compact but not properCompact Metric Space as a Countable Union of Closed SetsIf every closed ball in a metric space $X$ is compact, show that $X$ is separable.Locally compact Stone duality: Can a boolean space be recovered from its boolean algebra of clopen sets alone?Countably compact metric space is compactAre second-countable metric spaces $sigma$-compact?
$begingroup$
Let $(X, d)$ be a compact, metric space. Then in X there is at most countable amount of clopen sets. Hint: Compact metric spaces are separable.
That was an optional, harder problem to solve on my introductory topology course exam, which I had no idea how to prove.
general-topology
edited 1 hour ago
bof
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
$endgroup$
add a comment |
$begingroup$
Let $(X, d)$ be a compact, metric space. Then in X there is at most countable amount of clopen sets. Hint: Compact metric spaces are separable.
That was an optional, harder problem to solve on my introductory topology course exam, which I had no idea how to prove.
general-topology
edited 1 hour ago
bof
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
$endgroup$
$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago
add a comment |
$begingroup$
Let $(X, d)$ be a compact, metric space. Then in X there is at most countable amount of clopen sets. Hint: Compact metric spaces are separable.
That was an optional, harder problem to solve on my introductory topology course exam, which I had no idea how to prove.
general-topology
edited 1 hour ago
bof
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
$endgroup$
Let $(X, d)$ be a compact, metric space. Then in X there is at most countable amount of clopen sets. Hint: Compact metric spaces are separable.
That was an optional, harder problem to solve on my introductory topology course exam, which I had no idea how to prove.
general-topology
general-topology
edited 1 hour ago
bof
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
edited 1 hour ago
bof
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
edited 1 hour ago
bof
53.1k559121
edited 1 hour ago
bof
53.1k559121
edited 1 hour ago
bof
53.1k559121
53.1k559121
asked 3 hours ago
math_beginnermath_beginner
1848
asked 3 hours ago
math_beginnermath_beginner
1848
asked 3 hours ago
math_beginnermath_beginner
1848
1848
$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago
add a comment |
$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago
$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago
$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is false in general (it holds in metric spaces though).
General counterexample: $0,1^I$ in the product topology has $|I|$ many clopen sets (all sets of the form $pi_i^-1[j]$ for $i in I, j=0,1$ for starters).
In metric spaces, let $mathcalB$ be a countable base for $X$ (any compact metric space is separable and hence has a countable base). If $C$ is clopen, for each $x in C$ we can pick $B_x$ in the base such that $x in B_x subseteq C$ by openness of $C$. $C$ is also closed and hence compact so finitely many basic element also cover $C$.
So every clopen $C$ we can write as a finite union of basic sets from a countable base. There are only countably many such finite unions.
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
$endgroup$
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
This is false in general (it holds in metric spaces though).
General counterexample: $0,1^I$ in the product topology has $|I|$ many clopen sets (all sets of the form $pi_i^-1[j]$ for $i in I, j=0,1$ for starters).
In metric spaces, let $mathcalB$ be a countable base for $X$ (any compact metric space is separable and hence has a countable base). If $C$ is clopen, for each $x in C$ we can pick $B_x$ in the base such that $x in B_x subseteq C$ by openness of $C$. $C$ is also closed and hence compact so finitely many basic element also cover $C$.
So every clopen $C$ we can write as a finite union of basic sets from a countable base. There are only countably many such finite unions.
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
$endgroup$
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
add a comment |
$begingroup$
This is false in general (it holds in metric spaces though).
General counterexample: $0,1^I$ in the product topology has $|I|$ many clopen sets (all sets of the form $pi_i^-1[j]$ for $i in I, j=0,1$ for starters).
In metric spaces, let $mathcalB$ be a countable base for $X$ (any compact metric space is separable and hence has a countable base). If $C$ is clopen, for each $x in C$ we can pick $B_x$ in the base such that $x in B_x subseteq C$ by openness of $C$. $C$ is also closed and hence compact so finitely many basic element also cover $C$.
So every clopen $C$ we can write as a finite union of basic sets from a countable base. There are only countably many such finite unions.
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
$endgroup$
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
add a comment |
$begingroup$
This is false in general (it holds in metric spaces though).
General counterexample: $0,1^I$ in the product topology has $|I|$ many clopen sets (all sets of the form $pi_i^-1[j]$ for $i in I, j=0,1$ for starters).
In metric spaces, let $mathcalB$ be a countable base for $X$ (any compact metric space is separable and hence has a countable base). If $C$ is clopen, for each $x in C$ we can pick $B_x$ in the base such that $x in B_x subseteq C$ by openness of $C$. $C$ is also closed and hence compact so finitely many basic element also cover $C$.
So every clopen $C$ we can write as a finite union of basic sets from a countable base. There are only countably many such finite unions.
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
$endgroup$
This is false in general (it holds in metric spaces though).
General counterexample: $0,1^I$ in the product topology has $|I|$ many clopen sets (all sets of the form $pi_i^-1[j]$ for $i in I, j=0,1$ for starters).
In metric spaces, let $mathcalB$ be a countable base for $X$ (any compact metric space is separable and hence has a countable base). If $C$ is clopen, for each $x in C$ we can pick $B_x$ in the base such that $x in B_x subseteq C$ by openness of $C$. $C$ is also closed and hence compact so finitely many basic element also cover $C$.
So every clopen $C$ we can write as a finite union of basic sets from a countable base. There are only countably many such finite unions.
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
edited 2 hours ago
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
answered 2 hours ago
Henno BrandsmaHenno Brandsma
119k351130
119k351130
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
add a comment |
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
My fault - X was indeed a metric space. I have edited my question.
$endgroup$
– math_beginner
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
$begingroup$
@math_beginner I expanded my answer.
$endgroup$
– Henno Brandsma
2 hours ago
add a comment |
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$begingroup$
There are non-separable compact spaces.
$endgroup$
– José Carlos Santos
2 hours ago