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Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topological analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that this type of perspective/coding makes tradmark tools of the trade, like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.

Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(S, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topological analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that this type of perspective/coding makes tradmark tools of the trade, like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.

Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topological analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that type of perspective/coding makes tradmark tools of the trade like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.

Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topological analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that this type of perspective/coding makes tradmark tools of the trade, like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.

Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topologically flavored analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that type of perspective/coding makes tradmark tools of the trade like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.

Reasonable? Not really.

Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $\mathsf{CH}$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This makes the idea that there is some useful topological characterization of proper unlikely.

That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, making them comfortably embedded as a subspace of some product space $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.) Moreover, the exploitation of such codings/embeddings/etc.. is implicit in topological analysis of forcing extensions and motivates axioms like the various flavors of $\mathsf{OCA}$.

Anecdotally, I find that type of perspective/coding makes tradmark tools of the trade like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.