Do any topology textbooks start from the closed set definition?
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I am very interested in closed-set topologies, in particular because the induced topology on a manifold can be defined in terms of the smooth-scalar-field structure on that manifold as “a set of points in the manifold is closed when it is the kernel of a smooth scalar field.” This definition has kernel union as scalar multiplication and kernel intersection as a sort of Euclidean squared-sum of scalar fields; infinite unions are obviously dangerous (no constituent field maps this point to zero, but the infinite product does) while infinite intersections are obviously not (if the infinite squared sum maps this point to zero, all of the constituent points must have, too). Since the term “smooth” here means “closed under functions $C^infty(mathbb R^k, mathbb R)$ applied pointwise across any $k$-tuples of smooth scalar fields, for all $k$,” we have bump functions and it is very easy to use the closed-set definition of continuity to see that all of the smooth scalar fields are continuous on this topology.
Now I have noticed that every topology textbook I have picked up prefers to start with open sets and neighborhoods, in which case one needs to introduce those ideas and “think backwards” in this context before things “come back together.”
Can you recommend resources that start from the closed set definition and therefore hang fewer of their proofs on open sets and neighborhoods?
general-topology manifolds book-recommendation
asked Dec 2 at 14:42
CR Drost
1,810711
add a comment |
up vote
2
down vote
favorite
I am very interested in closed-set topologies, in particular because the induced topology on a manifold can be defined in terms of the smooth-scalar-field structure on that manifold as “a set of points in the manifold is closed when it is the kernel of a smooth scalar field.” This definition has kernel union as scalar multiplication and kernel intersection as a sort of Euclidean squared-sum of scalar fields; infinite unions are obviously dangerous (no constituent field maps this point to zero, but the infinite product does) while infinite intersections are obviously not (if the infinite squared sum maps this point to zero, all of the constituent points must have, too). Since the term “smooth” here means “closed under functions $C^infty(mathbb R^k, mathbb R)$ applied pointwise across any $k$-tuples of smooth scalar fields, for all $k$,” we have bump functions and it is very easy to use the closed-set definition of continuity to see that all of the smooth scalar fields are continuous on this topology.
Now I have noticed that every topology textbook I have picked up prefers to start with open sets and neighborhoods, in which case one needs to introduce those ideas and “think backwards” in this context before things “come back together.”
Can you recommend resources that start from the closed set definition and therefore hang fewer of their proofs on open sets and neighborhoods?
general-topology manifolds book-recommendation
asked Dec 2 at 14:42
CR Drost
1,810711
1
The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am very interested in closed-set topologies, in particular because the induced topology on a manifold can be defined in terms of the smooth-scalar-field structure on that manifold as “a set of points in the manifold is closed when it is the kernel of a smooth scalar field.” This definition has kernel union as scalar multiplication and kernel intersection as a sort of Euclidean squared-sum of scalar fields; infinite unions are obviously dangerous (no constituent field maps this point to zero, but the infinite product does) while infinite intersections are obviously not (if the infinite squared sum maps this point to zero, all of the constituent points must have, too). Since the term “smooth” here means “closed under functions $C^infty(mathbb R^k, mathbb R)$ applied pointwise across any $k$-tuples of smooth scalar fields, for all $k$,” we have bump functions and it is very easy to use the closed-set definition of continuity to see that all of the smooth scalar fields are continuous on this topology.
Now I have noticed that every topology textbook I have picked up prefers to start with open sets and neighborhoods, in which case one needs to introduce those ideas and “think backwards” in this context before things “come back together.”
Can you recommend resources that start from the closed set definition and therefore hang fewer of their proofs on open sets and neighborhoods?
general-topology manifolds book-recommendation
asked Dec 2 at 14:42
CR Drost
1,810711
I am very interested in closed-set topologies, in particular because the induced topology on a manifold can be defined in terms of the smooth-scalar-field structure on that manifold as “a set of points in the manifold is closed when it is the kernel of a smooth scalar field.” This definition has kernel union as scalar multiplication and kernel intersection as a sort of Euclidean squared-sum of scalar fields; infinite unions are obviously dangerous (no constituent field maps this point to zero, but the infinite product does) while infinite intersections are obviously not (if the infinite squared sum maps this point to zero, all of the constituent points must have, too). Since the term “smooth” here means “closed under functions $C^infty(mathbb R^k, mathbb R)$ applied pointwise across any $k$-tuples of smooth scalar fields, for all $k$,” we have bump functions and it is very easy to use the closed-set definition of continuity to see that all of the smooth scalar fields are continuous on this topology.
Now I have noticed that every topology textbook I have picked up prefers to start with open sets and neighborhoods, in which case one needs to introduce those ideas and “think backwards” in this context before things “come back together.”
Can you recommend resources that start from the closed set definition and therefore hang fewer of their proofs on open sets and neighborhoods?
general-topology manifolds book-recommendation
general-topology manifolds book-recommendation
asked Dec 2 at 14:42
CR Drost
1,810711
asked Dec 2 at 14:42
CR Drost
1,810711
edited Dec 2 at 14:50
asked Dec 2 at 14:42
CR Drost
1,810711
asked Dec 2 at 14:42
CR Drost
1,810711
asked Dec 2 at 14:42
CR Drost
1,810711
1,810711
1
The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18
add a comment |
1
The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18
1
1
The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18
The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18
add a comment |
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The definitions are trivially equivalent (de Morgan) and many books will define some concepts in both forms when they're introduced, like compactness and connectedness. Separation axioms and many covering axioms are less natural in terms of closed set though. Convergence too is less natural, I think. E.g. define $x_n to x$ without mentioning open sets... It's nice to have alternative forms sometimes, but no book can avoid mentioning open sets and neighbourhoods.
– Henno Brandsma
Dec 2 at 15:18