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It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the premise the question makes that algebraic vector spaces are any more basic.

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the assumption the question makes that algebraic vector spaces are any more basic than topological vector spaces.

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the premise that algebraic vector spaces are more basic.

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the premise the question makes that algebraic vector spaces are any more basic.

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom is usually at least separable, I think? Anyway, this rules out many topologies on vector spaces of infinite dimension.

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the premise that algebraic vector spaces are more basic.