Translating the definition of a basis in topology to subsidary deduction rules
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I am trying to write the definition of a basis for a topological space as a subsidiary deduction rule. A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $(forall x in X:exists B in beta: x in B) wedge (forall x in X:forall B_1 in beta:forall B_2 in beta:x in B_1 cap B_2 rightarrow exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2)$.
My question is that is it appropriate to write the former definition in the following form.
A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $vdash forall x in X:exists B in beta: x in B$ and $x in X,B_1 in beta, B_2 in beta, x in B_1 cap B_2 vdash exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2$?
My confusion is owing to the fact that the property of being a basis appears to be a semantic one (I am not sure) and given that, if it is appropriate to use $vdash$ instead of $vDash$. Secondly, since the subsidiary deductions are true contingently, depending on if $beta$ is a basis or not, is it acceptable to use them. What would then be an appropriate way to write the definition in a metalanguage other than natural language?
general-topology logic
asked 2 days ago
Anirban Mandal
796
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I am trying to write the definition of a basis for a topological space as a subsidiary deduction rule. A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $(forall x in X:exists B in beta: x in B) wedge (forall x in X:forall B_1 in beta:forall B_2 in beta:x in B_1 cap B_2 rightarrow exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2)$.
My question is that is it appropriate to write the former definition in the following form.
A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $vdash forall x in X:exists B in beta: x in B$ and $x in X,B_1 in beta, B_2 in beta, x in B_1 cap B_2 vdash exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2$?
My confusion is owing to the fact that the property of being a basis appears to be a semantic one (I am not sure) and given that, if it is appropriate to use $vdash$ instead of $vDash$. Secondly, since the subsidiary deductions are true contingently, depending on if $beta$ is a basis or not, is it acceptable to use them. What would then be an appropriate way to write the definition in a metalanguage other than natural language?
general-topology logic
asked 2 days ago
Anirban Mandal
796
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to write the definition of a basis for a topological space as a subsidiary deduction rule. A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $(forall x in X:exists B in beta: x in B) wedge (forall x in X:forall B_1 in beta:forall B_2 in beta:x in B_1 cap B_2 rightarrow exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2)$.
My question is that is it appropriate to write the former definition in the following form.
A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $vdash forall x in X:exists B in beta: x in B$ and $x in X,B_1 in beta, B_2 in beta, x in B_1 cap B_2 vdash exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2$?
My confusion is owing to the fact that the property of being a basis appears to be a semantic one (I am not sure) and given that, if it is appropriate to use $vdash$ instead of $vDash$. Secondly, since the subsidiary deductions are true contingently, depending on if $beta$ is a basis or not, is it acceptable to use them. What would then be an appropriate way to write the definition in a metalanguage other than natural language?
general-topology logic
asked 2 days ago
Anirban Mandal
796
I am trying to write the definition of a basis for a topological space as a subsidiary deduction rule. A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $(forall x in X:exists B in beta: x in B) wedge (forall x in X:forall B_1 in beta:forall B_2 in beta:x in B_1 cap B_2 rightarrow exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2)$.
My question is that is it appropriate to write the former definition in the following form.
A collection $beta$ of subsets of a set $X$ is a basis for a topology on $X$ if and only if $vdash forall x in X:exists B in beta: x in B$ and $x in X,B_1 in beta, B_2 in beta, x in B_1 cap B_2 vdash exists B_3 in beta:x in B_3 wedge B_3 subset B_1 cap B_2$?
My confusion is owing to the fact that the property of being a basis appears to be a semantic one (I am not sure) and given that, if it is appropriate to use $vdash$ instead of $vDash$. Secondly, since the subsidiary deductions are true contingently, depending on if $beta$ is a basis or not, is it acceptable to use them. What would then be an appropriate way to write the definition in a metalanguage other than natural language?
general-topology logic
general-topology logic
asked 2 days ago
Anirban Mandal
796
asked 2 days ago
Anirban Mandal
796
edited yesterday
asked 2 days ago
Anirban Mandal
796
asked 2 days ago
Anirban Mandal
796
asked 2 days ago
Anirban Mandal
796
796
add a comment |
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