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et qui - how do you really understand that kind of phraseology?
Non-trivial topology where only open sets are closed
Exhaustion of open sets by closed setsNatural non-trivial topology on $mathbb R$ such that there are more than $2^mathbb N$ open setswhy use open interval rather than closed interval as open sets for real line topologyA non-trivial Hausdorff topology on XClosed sets in Noetherian topologyIn the finite complement topology on $mathbbR$, is the subset $ x $ closed?Are closed sets in topology?For which topologies are the concepts of open and closed “interchangeable”?How to determine which sets are open in a topology?Inclusion of open sets in closed sets of Zariski topology
$begingroup$
For example, on $mathbbR$ there exists trivial topology which contains only $mathbbR$ and $emptyset$ and in that topology all open sets are closed and all closed sets are open.
Question. Does there exist non-trivial topology on $mathbbR$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?
I hope my question is not meaningless.
Thank you for any help.
general-topology
$endgroup$
add a comment |
$begingroup$
For example, on $mathbbR$ there exists trivial topology which contains only $mathbbR$ and $emptyset$ and in that topology all open sets are closed and all closed sets are open.
Question. Does there exist non-trivial topology on $mathbbR$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?
I hope my question is not meaningless.
Thank you for any help.
general-topology
$endgroup$
2
$begingroup$
The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
$endgroup$
– parsiad
4 hours ago
1
$begingroup$
(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
$endgroup$
– Arturo Magidin
4 hours ago
1
$begingroup$
Related to your question are door spaces and extremally disconnected spaces.
$endgroup$
– William Elliot
4 hours ago
add a comment |
$begingroup$
For example, on $mathbbR$ there exists trivial topology which contains only $mathbbR$ and $emptyset$ and in that topology all open sets are closed and all closed sets are open.
Question. Does there exist non-trivial topology on $mathbbR$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?
I hope my question is not meaningless.
Thank you for any help.
general-topology
$endgroup$
For example, on $mathbbR$ there exists trivial topology which contains only $mathbbR$ and $emptyset$ and in that topology all open sets are closed and all closed sets are open.
Question. Does there exist non-trivial topology on $mathbbR$ for which all open sets are closed and all closed sets are open? Further, given some general set $X$ whose number of elements is finite, could we always construct non-trivial topology where all open sets are closed and all closed sets are open? What if $X$ has a non-finite number of elements?
I hope my question is not meaningless.
Thank you for any help.
general-topology
general-topology
edited 2 hours ago
YuiTo Cheng
2,0532637
2,0532637
asked 5 hours ago
ThomThom
346111
346111
2
$begingroup$
The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
$endgroup$
– parsiad
4 hours ago
1
$begingroup$
(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
$endgroup$
– Arturo Magidin
4 hours ago
1
$begingroup$
Related to your question are door spaces and extremally disconnected spaces.
$endgroup$
– William Elliot
4 hours ago
add a comment |
2
$begingroup$
The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
$endgroup$
– parsiad
4 hours ago
1
$begingroup$
(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
$endgroup$
– Arturo Magidin
4 hours ago
1
$begingroup$
Related to your question are door spaces and extremally disconnected spaces.
$endgroup$
– William Elliot
4 hours ago
2
2
$begingroup$
The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
$endgroup$
– parsiad
4 hours ago
$begingroup$
The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
$endgroup$
– parsiad
4 hours ago
1
1
$begingroup$
(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
$endgroup$
– Arturo Magidin
4 hours ago
1
1
$begingroup$
Related to your question are door spaces and extremally disconnected spaces.
$endgroup$
– William Elliot
4 hours ago
$begingroup$
Related to your question are door spaces and extremally disconnected spaces.
$endgroup$
– William Elliot
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.
Specifically, let $mathcalP=X_i_iin I$ be a partition of $X$, and let $tau$ be the collection of sets of the form $cup_iin I_0X_i$ for $I_0subseteq I$. Then $tau$ is a topology: the empty set corresponds to $I_0=varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $tau$ has the desired property.
Now let $tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $xsim y$ if and only if for every $Ain tau$, $xin A$ if and only if $yin A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $tau$ is in fact, the topology induced by this partition as above.
Indeed, if $Ain tau$, then $A=cup_xin A[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $yin[x]$, then since $xin A$ then $yin A$, so we have equality.
Now, conversely, let $xin X$ and look at $[x]$. I claim that $X-[x]$ lies in $tau$. So see this, let $zin X-[x]$. Then since $znotin [x]$, there exists an open set $A_zin tau$ such that $zin A_z$ but $xnotin A_z$ (and hence, $[x]cap A_z=varnothing$). Now, $cup_znotin[x]A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]intau$.
We have then proven that every element of the partition induced by $sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/sim$.
$endgroup$
add a comment |
$begingroup$
You can manufacture examples like this pretty easily. Let $A$ be any subset of $mathbbR$. Put a topology on $mathbbR$ with the following open sets:
$$
varnothing, A, mathbbR-A, mathbbR.
$$
You can easily check that this always gives a topology, and a subset of $mathbbR$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $mathbbR$ here.
$endgroup$
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
$endgroup$
– Thom
4 hours ago
$begingroup$
@Thom: Yes. I’ll write it up.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Fascinating. Thank you.
$endgroup$
– Thom
4 hours ago
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.
Specifically, let $mathcalP=X_i_iin I$ be a partition of $X$, and let $tau$ be the collection of sets of the form $cup_iin I_0X_i$ for $I_0subseteq I$. Then $tau$ is a topology: the empty set corresponds to $I_0=varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $tau$ has the desired property.
Now let $tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $xsim y$ if and only if for every $Ain tau$, $xin A$ if and only if $yin A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $tau$ is in fact, the topology induced by this partition as above.
Indeed, if $Ain tau$, then $A=cup_xin A[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $yin[x]$, then since $xin A$ then $yin A$, so we have equality.
Now, conversely, let $xin X$ and look at $[x]$. I claim that $X-[x]$ lies in $tau$. So see this, let $zin X-[x]$. Then since $znotin [x]$, there exists an open set $A_zin tau$ such that $zin A_z$ but $xnotin A_z$ (and hence, $[x]cap A_z=varnothing$). Now, $cup_znotin[x]A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]intau$.
We have then proven that every element of the partition induced by $sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/sim$.
$endgroup$
add a comment |
$begingroup$
The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.
Specifically, let $mathcalP=X_i_iin I$ be a partition of $X$, and let $tau$ be the collection of sets of the form $cup_iin I_0X_i$ for $I_0subseteq I$. Then $tau$ is a topology: the empty set corresponds to $I_0=varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $tau$ has the desired property.
Now let $tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $xsim y$ if and only if for every $Ain tau$, $xin A$ if and only if $yin A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $tau$ is in fact, the topology induced by this partition as above.
Indeed, if $Ain tau$, then $A=cup_xin A[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $yin[x]$, then since $xin A$ then $yin A$, so we have equality.
Now, conversely, let $xin X$ and look at $[x]$. I claim that $X-[x]$ lies in $tau$. So see this, let $zin X-[x]$. Then since $znotin [x]$, there exists an open set $A_zin tau$ such that $zin A_z$ but $xnotin A_z$ (and hence, $[x]cap A_z=varnothing$). Now, $cup_znotin[x]A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]intau$.
We have then proven that every element of the partition induced by $sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/sim$.
$endgroup$
add a comment |
$begingroup$
The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.
Specifically, let $mathcalP=X_i_iin I$ be a partition of $X$, and let $tau$ be the collection of sets of the form $cup_iin I_0X_i$ for $I_0subseteq I$. Then $tau$ is a topology: the empty set corresponds to $I_0=varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $tau$ has the desired property.
Now let $tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $xsim y$ if and only if for every $Ain tau$, $xin A$ if and only if $yin A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $tau$ is in fact, the topology induced by this partition as above.
Indeed, if $Ain tau$, then $A=cup_xin A[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $yin[x]$, then since $xin A$ then $yin A$, so we have equality.
Now, conversely, let $xin X$ and look at $[x]$. I claim that $X-[x]$ lies in $tau$. So see this, let $zin X-[x]$. Then since $znotin [x]$, there exists an open set $A_zin tau$ such that $zin A_z$ but $xnotin A_z$ (and hence, $[x]cap A_z=varnothing$). Now, $cup_znotin[x]A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]intau$.
We have then proven that every element of the partition induced by $sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/sim$.
$endgroup$
The topologies with this property are precisely the ones that are derived from partitions of the underlying set $X$.
Specifically, let $mathcalP=X_i_iin I$ be a partition of $X$, and let $tau$ be the collection of sets of the form $cup_iin I_0X_i$ for $I_0subseteq I$. Then $tau$ is a topology: the empty set corresponds to $I_0=varnothing$, the set $X$ to $I_0=I$; the union of such sets corresponds to the family indexed by the union of indices, and the intersection to the intersection of the indices. Moreover, the complement of the set corresponding to $I_0$ is the set corresponding to $I-I_0$. Thus, $tau$ has the desired property.
Now let $tau$ be any topology on $X$ with the desired property. Define an equivalence relation on $X$ by letting $xsim y$ if and only if for every $Ain tau$, $xin A$ if and only if $yin A$. Trivially, this is an equivalence relation, and so induces a partition on $X$. I claim that $tau$ is in fact, the topology induced by this partition as above.
Indeed, if $Ain tau$, then $A=cup_xin A[x]$, where $[x]$ is the equivalence class of $x$. Trivially $A$ is contained in the right hand side, and if $yin[x]$, then since $xin A$ then $yin A$, so we have equality.
Now, conversely, let $xin X$ and look at $[x]$. I claim that $X-[x]$ lies in $tau$. So see this, let $zin X-[x]$. Then since $znotin [x]$, there exists an open set $A_zin tau$ such that $zin A_z$ but $xnotin A_z$ (and hence, $[x]cap A_z=varnothing$). Now, $cup_znotin[x]A_z$ is an open set, contains every element of $X-[x]$, and intersects $[x]$ trivially because each element in the union does. That is, this open set is $A-[x]$; but since the complement of every open set is open, and $A-[x]$ is open, then $[x]$ is open. Thus, $[x]intau$.
We have then proven that every element of the partition induced by $sim$ is open, and that every open set is a union of such elements of the partition. That is, the open sets are precisely the unions of elements of the partition $X/sim$.
answered 4 hours ago
Arturo MagidinArturo Magidin
265k34590918
265k34590918
add a comment |
add a comment |
$begingroup$
You can manufacture examples like this pretty easily. Let $A$ be any subset of $mathbbR$. Put a topology on $mathbbR$ with the following open sets:
$$
varnothing, A, mathbbR-A, mathbbR.
$$
You can easily check that this always gives a topology, and a subset of $mathbbR$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $mathbbR$ here.
$endgroup$
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
$endgroup$
– Thom
4 hours ago
$begingroup$
@Thom: Yes. I’ll write it up.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Fascinating. Thank you.
$endgroup$
– Thom
4 hours ago
add a comment |
$begingroup$
You can manufacture examples like this pretty easily. Let $A$ be any subset of $mathbbR$. Put a topology on $mathbbR$ with the following open sets:
$$
varnothing, A, mathbbR-A, mathbbR.
$$
You can easily check that this always gives a topology, and a subset of $mathbbR$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $mathbbR$ here.
$endgroup$
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
$endgroup$
– Thom
4 hours ago
$begingroup$
@Thom: Yes. I’ll write it up.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Fascinating. Thank you.
$endgroup$
– Thom
4 hours ago
add a comment |
$begingroup$
You can manufacture examples like this pretty easily. Let $A$ be any subset of $mathbbR$. Put a topology on $mathbbR$ with the following open sets:
$$
varnothing, A, mathbbR-A, mathbbR.
$$
You can easily check that this always gives a topology, and a subset of $mathbbR$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $mathbbR$ here.
$endgroup$
You can manufacture examples like this pretty easily. Let $A$ be any subset of $mathbbR$. Put a topology on $mathbbR$ with the following open sets:
$$
varnothing, A, mathbbR-A, mathbbR.
$$
You can easily check that this always gives a topology, and a subset of $mathbbR$ is open if and only if it is closed. This construction generalizes to any set $X$: there's nothing special about $mathbbR$ here.
answered 4 hours ago
RandallRandall
10.6k11431
10.6k11431
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
$endgroup$
– Thom
4 hours ago
$begingroup$
@Thom: Yes. I’ll write it up.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Fascinating. Thank you.
$endgroup$
– Thom
4 hours ago
add a comment |
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
$endgroup$
– Thom
4 hours ago
$begingroup$
@Thom: Yes. I’ll write it up.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
@ArturoMagidin Fascinating. Thank you.
$endgroup$
– Thom
4 hours ago
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
$endgroup$
– Arturo Magidin
4 hours ago
$begingroup$
More generally, take any partition and take unions of elements of the partition. This example is what you get from the partition $A,mathbbR-A$.
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– Arturo Magidin
4 hours ago
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@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
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– Thom
4 hours ago
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@ArturoMagidin Let us denote with X set of all topologies which satisfy property in the question on $mathbbR$. Can every topology in $X$ be made with some partitions?
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– Thom
4 hours ago
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@Thom: Yes. I’ll write it up.
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– Arturo Magidin
4 hours ago
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@Thom: Yes. I’ll write it up.
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– Arturo Magidin
4 hours ago
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@ArturoMagidin Fascinating. Thank you.
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– Thom
4 hours ago
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@ArturoMagidin Fascinating. Thank you.
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– Thom
4 hours ago
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2
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The trivial topology on a set $X$ is $emptyset, X$. The discrete topology on a set $X$ is $2^X$. Both have the property that you are looking for, though I am not sure if you are using the term "trivial" here in reference to the trivial topology or just to mean "simple".
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– parsiad
4 hours ago
1
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(1) The discrete topology, where all subsets are open and closed. (2) Given any partition $X_i_iin Iof $X$, take as a topology the colleciton of all sets that are unions of elements of the partition; the complements are also unions of elements of the partition. This works for any set, regardless of cardinality, and any partition, regarless of size of the partition or of its constituent sets.
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– Arturo Magidin
4 hours ago
1
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Related to your question are door spaces and extremally disconnected spaces.
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– William Elliot
4 hours ago