Is every funky topology a topology?

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Whenever $X$ is a set, define that a funky topology on $X$ is a collection of subsets of $X$ deemed "open" such that:

$X$ is open.

If $A,B subseteq X$ is open, then $A cap B$ is open.

If $mathcalC$ is an open cover of $X$, then for all $A subseteq X$, if $$forall U in mathcalC(A cap U mbox is open),$$ then $A$ is open.

It's clear that every topology is a funky topology.

Question. Does the converse hold?

asked Sep 9 at 2:58

goblin

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up vote
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Whenever $X$ is a set, define that a funky topology on $X$ is a collection of subsets of $X$ deemed "open" such that:

$X$ is open.

If $A,B subseteq X$ is open, then $A cap B$ is open.

If $mathcalC$ is an open cover of $X$, then for all $A subseteq X$, if $$forall U in mathcalC(A cap U mbox is open),$$ then $A$ is open.

It's clear that every topology is a funky topology.

Question. Does the converse hold?

asked Sep 9 at 2:58

goblin

35.8k1154182

add a commentÂ |Â

up vote
4
down vote

favorite

Whenever $X$ is a set, define that a funky topology on $X$ is a collection of subsets of $X$ deemed "open" such that:

$X$ is open.

If $A,B subseteq X$ is open, then $A cap B$ is open.

If $mathcalC$ is an open cover of $X$, then for all $A subseteq X$, if $$forall U in mathcalC(A cap U mbox is open),$$ then $A$ is open.

It's clear that every topology is a funky topology.

Question. Does the converse hold?

asked Sep 9 at 2:58

goblin

35.8k1154182

Whenever $X$ is a set, define that a funky topology on $X$ is a collection of subsets of $X$ deemed "open" such that:

$X$ is open.

If $A,B subseteq X$ is open, then $A cap B$ is open.

If $mathcalC$ is an open cover of $X$, then for all $A subseteq X$, if $$forall U in mathcalC(A cap U mbox is open),$$ then $A$ is open.

It's clear that every topology is a funky topology.

Question. Does the converse hold?

general-topology

asked Sep 9 at 2:58

goblin

35.8k1154182

asked Sep 9 at 2:58

goblin

35.8k1154182

asked Sep 9 at 2:58

goblin

35.8k1154182

asked Sep 9 at 2:58

goblin

35.8k1154182

asked Sep 9 at 2:58

goblin

35.8k1154182

add a commentÂ |Â

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No. For instance, $X$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=a,b,c$, then $emptyset,a,b,X$ is a funky topology but not a topology.

answered Sep 9 at 3:06

Eric Wofsey

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1 Answer
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active

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1 Answer
1

active

oldest

votes

up vote
5
down vote

accepted

No. For instance, $X$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=a,b,c$, then $emptyset,a,b,X$ is a funky topology but not a topology.

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

add a commentÂ |Â

up vote
5
down vote

accepted

No. For instance, $X$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=a,b,c$, then $emptyset,a,b,X$ is a funky topology but not a topology.

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

add a commentÂ |Â

up vote
5
down vote

accepted

No. For instance, $X$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=a,b,c$, then $emptyset,a,b,X$ is a funky topology but not a topology.

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

No. For instance, $X$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=a,b,c$, then $emptyset,a,b,X$ is a funky topology but not a topology.

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

answered Sep 9 at 3:06

Eric Wofsey

166k12195308

add a commentÂ |Â

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