Mixing
Is every finite, nondiscrete $T_0$ space connected?
$begingroup$
Is every finite, nondiscrete $T_0$ space connected?
What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost in it. It seems so "picky".
general-topology connectedness
edited Jan 30 at 12:03
Jean Marie
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
$endgroup$
add a comment |
$begingroup$
Is every finite, nondiscrete $T_0$ space connected?
What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost in it. It seems so "picky".
general-topology connectedness
edited Jan 30 at 12:03
Jean Marie
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
$endgroup$
2
$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
2
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
2
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08
add a comment |
$begingroup$
Is every finite, nondiscrete $T_0$ space connected?
What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost in it. It seems so "picky".
general-topology connectedness
edited Jan 30 at 12:03
Jean Marie
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
$endgroup$
Is every finite, nondiscrete $T_0$ space connected?
What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost in it. It seems so "picky".
general-topology connectedness
general-topology connectedness
edited Jan 30 at 12:03
Jean Marie
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
edited Jan 30 at 12:03
Jean Marie
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
edited Jan 30 at 12:03
Jean Marie
31.3k42255
edited Jan 30 at 12:03
Jean Marie
31.3k42255
edited Jan 30 at 12:03
Jean Marie
31.3k42255
31.3k42255
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
asked Jan 30 at 11:59
Nemanja BericNemanja Beric
37018
37018
2
$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
2
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
2
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08
add a comment |
2
$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
2
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
2
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08
2
2
$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
2
2
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
2
2
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08
add a comment |
1 Answer
1
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$begingroup$
Let $X={1,2,3}$ be endowed with topology $tau={varnothing,{2},{3},{1,2},{2,3},{1,2,3}}$.
Then $X$ is $T_0$ and e.g. ${3}$ is a non-trivial clopen set.
answered Jan 30 at 12:37
drhabdrhab
104k545136
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Let $X={1,2,3}$ be endowed with topology $tau={varnothing,{2},{3},{1,2},{2,3},{1,2,3}}$.
Then $X$ is $T_0$ and e.g. ${3}$ is a non-trivial clopen set.
answered Jan 30 at 12:37
drhabdrhab
104k545136
$endgroup$
add a comment |
$begingroup$
Let $X={1,2,3}$ be endowed with topology $tau={varnothing,{2},{3},{1,2},{2,3},{1,2,3}}$.
Then $X$ is $T_0$ and e.g. ${3}$ is a non-trivial clopen set.
answered Jan 30 at 12:37
drhabdrhab
104k545136
$endgroup$
add a comment |
$begingroup$
Let $X={1,2,3}$ be endowed with topology $tau={varnothing,{2},{3},{1,2},{2,3},{1,2,3}}$.
Then $X$ is $T_0$ and e.g. ${3}$ is a non-trivial clopen set.
answered Jan 30 at 12:37
drhabdrhab
104k545136
$endgroup$
Let $X={1,2,3}$ be endowed with topology $tau={varnothing,{2},{3},{1,2},{2,3},{1,2,3}}$.
Then $X$ is $T_0$ and e.g. ${3}$ is a non-trivial clopen set.
answered Jan 30 at 12:37
drhabdrhab
104k545136
answered Jan 30 at 12:37
drhabdrhab
104k545136
answered Jan 30 at 12:37
drhabdrhab
104k545136
answered Jan 30 at 12:37
drhabdrhab
104k545136
104k545136
add a comment |
add a comment |
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$begingroup$
what about $X={a,b,c}$ with the topology ${emptyset,{a,b},{b},{c},X}$? The space is not discrete, ${a,b}$ and ${c}$ separate the space, but it is $T_0$
$endgroup$
– Rylee Lyman
Jan 30 at 12:30
2
$begingroup$
@RyleeLyman You need ${b,c}$ too for a topology.
$endgroup$
– bof
Jan 30 at 14:04
2
$begingroup$
Take any finite nondiscrete $T_0$ and add an isolated point to it.
$endgroup$
– bof
Jan 30 at 14:08