What happens when every dense set is open?
$begingroup$
I am supposed to prove or disprove the following claim:
If in space $(X, mathcal{O})$ every dense set is open, then $(X, mathcal{O})$ is not $T_2$-space.
I tried taking arbitrary $x, y in X$ such that $x ne y$. But what now? No one guarantees me that every neighborhood $U$ of $x$ is dense, so it will intersect every neighborhood $V$ of $y$. On the other hand, to find a counterexample seems hard, because every finite $T_2$-space is discreet. Any help is appreciated.
general-topology
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
$endgroup$
add a comment |
$begingroup$
I am supposed to prove or disprove the following claim:
If in space $(X, mathcal{O})$ every dense set is open, then $(X, mathcal{O})$ is not $T_2$-space.
I tried taking arbitrary $x, y in X$ such that $x ne y$. But what now? No one guarantees me that every neighborhood $U$ of $x$ is dense, so it will intersect every neighborhood $V$ of $y$. On the other hand, to find a counterexample seems hard, because every finite $T_2$-space is discreet. Any help is appreciated.
general-topology
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
$endgroup$
add a comment |
$begingroup$
I am supposed to prove or disprove the following claim:
If in space $(X, mathcal{O})$ every dense set is open, then $(X, mathcal{O})$ is not $T_2$-space.
I tried taking arbitrary $x, y in X$ such that $x ne y$. But what now? No one guarantees me that every neighborhood $U$ of $x$ is dense, so it will intersect every neighborhood $V$ of $y$. On the other hand, to find a counterexample seems hard, because every finite $T_2$-space is discreet. Any help is appreciated.
general-topology
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
$endgroup$
I am supposed to prove or disprove the following claim:
If in space $(X, mathcal{O})$ every dense set is open, then $(X, mathcal{O})$ is not $T_2$-space.
I tried taking arbitrary $x, y in X$ such that $x ne y$. But what now? No one guarantees me that every neighborhood $U$ of $x$ is dense, so it will intersect every neighborhood $V$ of $y$. On the other hand, to find a counterexample seems hard, because every finite $T_2$-space is discreet. Any help is appreciated.
general-topology
general-topology
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
edited Dec 8 '18 at 9:47
Nemanja Beric
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
asked Dec 8 '18 at 8:57
Nemanja BericNemanja Beric
37018
37018
add a comment |
add a comment |
1 Answer
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Consider the discrete topology on a set $X$.
- Clearly $X$ is Hausdorff.
- The only dense subset of $X$ is $X$ itself, which is open. So every dense subset of $X$ is open.
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
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add a comment |
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$begingroup$
Consider the discrete topology on a set $X$.
- Clearly $X$ is Hausdorff.
- The only dense subset of $X$ is $X$ itself, which is open. So every dense subset of $X$ is open.
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
$endgroup$
add a comment |
$begingroup$
Consider the discrete topology on a set $X$.
- Clearly $X$ is Hausdorff.
- The only dense subset of $X$ is $X$ itself, which is open. So every dense subset of $X$ is open.
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
$endgroup$
add a comment |
$begingroup$
Consider the discrete topology on a set $X$.
- Clearly $X$ is Hausdorff.
- The only dense subset of $X$ is $X$ itself, which is open. So every dense subset of $X$ is open.
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
$endgroup$
Consider the discrete topology on a set $X$.
- Clearly $X$ is Hausdorff.
- The only dense subset of $X$ is $X$ itself, which is open. So every dense subset of $X$ is open.
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
answered Dec 8 '18 at 9:32
stochastic randomnessstochastic randomness
41017
41017
add a comment |
add a comment |
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