“Density” of Topological Properties.
I came up with this question when I drew diagrams to represent the relations among special topological space. I am aware that the answer might be trivial, but I haven’t figure out a way to do it yet.
Given two topological properties $P_1$ and $P_2$ such that $P_1<P_2$($P_1$ implies $P_2$, but does not equal to $P_2$. An example would be “$T_0<T_1$”), is there a topological property $P$ such that $P_1<P<P_2$?(“$<$” is defined as above) Can we define a topology using the class of topological properties?
general-topology
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
add a comment |
I came up with this question when I drew diagrams to represent the relations among special topological space. I am aware that the answer might be trivial, but I haven’t figure out a way to do it yet.
Given two topological properties $P_1$ and $P_2$ such that $P_1<P_2$($P_1$ implies $P_2$, but does not equal to $P_2$. An example would be “$T_0<T_1$”), is there a topological property $P$ such that $P_1<P<P_2$?(“$<$” is defined as above) Can we define a topology using the class of topological properties?
general-topology
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
add a comment |
I came up with this question when I drew diagrams to represent the relations among special topological space. I am aware that the answer might be trivial, but I haven’t figure out a way to do it yet.
Given two topological properties $P_1$ and $P_2$ such that $P_1<P_2$($P_1$ implies $P_2$, but does not equal to $P_2$. An example would be “$T_0<T_1$”), is there a topological property $P$ such that $P_1<P<P_2$?(“$<$” is defined as above) Can we define a topology using the class of topological properties?
general-topology
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
I came up with this question when I drew diagrams to represent the relations among special topological space. I am aware that the answer might be trivial, but I haven’t figure out a way to do it yet.
Given two topological properties $P_1$ and $P_2$ such that $P_1<P_2$($P_1$ implies $P_2$, but does not equal to $P_2$. An example would be “$T_0<T_1$”), is there a topological property $P$ such that $P_1<P<P_2$?(“$<$” is defined as above) Can we define a topology using the class of topological properties?
general-topology
general-topology
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
edited Dec 3 '18 at 23:34
Eric Wofsey
180k12207335
180k12207335
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
asked Dec 3 '18 at 19:59
William SunWilliam Sun
471111
471111
add a comment |
add a comment |
1 Answer
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The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.
So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $mathbb{R}$).
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
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active
oldest
votes
The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.
So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $mathbb{R}$).
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
add a comment |
The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.
So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $mathbb{R}$).
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
add a comment |
The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.
So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $mathbb{R}$).
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
The key thing to understand is that a topological property is really just a collection of homeomorphism classes of topological spaces. Given a collection $C$ of homeomorphism classes of topological spaces, the property of being in one of those homeomorphism classes is a topological property, and every topological property has this form.
So, how do you find two topological properties with nothing in between them? You just make them differ by exactly one homeomorphism class. For instance, let $P_2$ be the property of being Hausdorff, and let $P_1$ be the property of being Hausdorff but not homeomorphic to $mathbb{R}$. There is no topological property strictly in between these, since they differ only on a single homeomorphism class (namely that of $mathbb{R}$).
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
answered Dec 3 '18 at 23:37
Eric WofseyEric Wofsey
180k12207335
180k12207335
add a comment |
add a comment |
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