What are the closed sets of this topology.
$begingroup$
I have the topology consisting of these Sets.
$$X = {a,b,c,d,e,f}$$
$$O = {X,∅,{b},{c,d},{b,c,d},{a,c,d,e,f}}$$
The question is what are the closed sets of this topology?
I have found information on the 'closure' and 'interior' of sets, thought that that was not what was asked for and then stumbled across this.
Is that what I am supposed to do?
$$X^c = emptyset$$
$$emptyset^c = X$$
$${b}^c = {a, c, d, e, f}$$
$${c,d}^c= {a, b, e, f}$$
$${b, c, d}^c = {a, e, f}$$
$${a, c, d, e, f}^c = {b}$$
general-topology algebraic-topology
asked Dec 2 '18 at 23:32
thebillythebilly
566
$endgroup$
add a comment |
$begingroup$
I have the topology consisting of these Sets.
$$X = {a,b,c,d,e,f}$$
$$O = {X,∅,{b},{c,d},{b,c,d},{a,c,d,e,f}}$$
The question is what are the closed sets of this topology?
I have found information on the 'closure' and 'interior' of sets, thought that that was not what was asked for and then stumbled across this.
Is that what I am supposed to do?
$$X^c = emptyset$$
$$emptyset^c = X$$
$${b}^c = {a, c, d, e, f}$$
$${c,d}^c= {a, b, e, f}$$
$${b, c, d}^c = {a, e, f}$$
$${a, c, d, e, f}^c = {b}$$
general-topology algebraic-topology
asked Dec 2 '18 at 23:32
thebillythebilly
566
$endgroup$
$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15
add a comment |
$begingroup$
I have the topology consisting of these Sets.
$$X = {a,b,c,d,e,f}$$
$$O = {X,∅,{b},{c,d},{b,c,d},{a,c,d,e,f}}$$
The question is what are the closed sets of this topology?
I have found information on the 'closure' and 'interior' of sets, thought that that was not what was asked for and then stumbled across this.
Is that what I am supposed to do?
$$X^c = emptyset$$
$$emptyset^c = X$$
$${b}^c = {a, c, d, e, f}$$
$${c,d}^c= {a, b, e, f}$$
$${b, c, d}^c = {a, e, f}$$
$${a, c, d, e, f}^c = {b}$$
general-topology algebraic-topology
asked Dec 2 '18 at 23:32
thebillythebilly
566
$endgroup$
I have the topology consisting of these Sets.
$$X = {a,b,c,d,e,f}$$
$$O = {X,∅,{b},{c,d},{b,c,d},{a,c,d,e,f}}$$
The question is what are the closed sets of this topology?
I have found information on the 'closure' and 'interior' of sets, thought that that was not what was asked for and then stumbled across this.
Is that what I am supposed to do?
$$X^c = emptyset$$
$$emptyset^c = X$$
$${b}^c = {a, c, d, e, f}$$
$${c,d}^c= {a, b, e, f}$$
$${b, c, d}^c = {a, e, f}$$
$${a, c, d, e, f}^c = {b}$$
general-topology algebraic-topology
general-topology algebraic-topology
asked Dec 2 '18 at 23:32
thebillythebilly
566
asked Dec 2 '18 at 23:32
thebillythebilly
566
asked Dec 2 '18 at 23:32
thebillythebilly
566
asked Dec 2 '18 at 23:32
thebillythebilly
566
asked Dec 2 '18 at 23:32
thebillythebilly
566
566
$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15
add a comment |
$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15
add a comment |
1 Answer
1
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$begingroup$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C={∅,X,{b},{a,e,f},{a,b,e,f},{a,c,d,e,f}}$$ you will obtain another set already in $C$.
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
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$begingroup$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C={∅,X,{b},{a,e,f},{a,b,e,f},{a,c,d,e,f}}$$ you will obtain another set already in $C$.
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
$endgroup$
add a comment |
$begingroup$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C={∅,X,{b},{a,e,f},{a,b,e,f},{a,c,d,e,f}}$$ you will obtain another set already in $C$.
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
$endgroup$
add a comment |
$begingroup$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C={∅,X,{b},{a,e,f},{a,b,e,f},{a,c,d,e,f}}$$ you will obtain another set already in $C$.
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
$endgroup$
The closed sets of the topology are given by the complements of the sets in $O$, as well as finite unions and arbitrary intersections of these sets. In fact, in this case, just by taking complements as you did, you have found all closed sets in the topology. If you take a union or intersection of any two sets in $$C={∅,X,{b},{a,e,f},{a,b,e,f},{a,c,d,e,f}}$$ you will obtain another set already in $C$.
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
answered Dec 2 '18 at 23:47
greeliousgreelious
19410
19410
add a comment |
add a comment |
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$begingroup$
Also the infinite intersection finite union of closed sets are closed.
$endgroup$
– Joel Pereira
Dec 2 '18 at 23:36
$begingroup$
This is correct, assuming $O$ is the set of open sets in the topology.
$endgroup$
– platty
Dec 2 '18 at 23:37
$begingroup$
@JoelPereira So what do I need to add?
$endgroup$
– thebilly
Dec 8 '18 at 14:15
$begingroup$
@platty awesome. thank you.
$endgroup$
– thebilly
Dec 8 '18 at 14:15