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Minor Math Jaxing and formatting

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edited Sep 14, 2023 at 9:42

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I have posted this question on MSE one year ago here https://math.stackexchange.com/q/4368193/223051this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here.

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry.

Does it mean, we have a one parameter family $\omega_t$, $t\in [a,b]$ of symplectic forms? If in that case why the word continuity though?
Or does it mean we have a continuous map $f:[a,b]\to \Omega^2_{Symp}$$f:[a,b]\to \Omega^2_\text{Symp}$, where $\Omega^2_{Symp}$$\Omega^2_\text{Symp}$ is the space of all symplectic forms on an ambient manifold.? But in this case what is the topology of $\Omega^2_{Symp}$$\Omega^2_\text{Symp}$?

Can anyone please help clarify the definition for continuous family.

I have a strong sense that my second interpretation is the right one. But the thing is I don't know the topology of the space of differential forms $\Omega(M)$ on $M$. This might be a standard topology because I have seen in McDuff's book: Introduction to symplectic topology Introduction to symplectic topology, assumes that $\Omega^2(M)$ is a topological space.

I have posted this question on MSE one year ago here https://math.stackexchange.com/q/4368193/223051 , but till now I did not received an answer. Therefore I have decided to post it here.

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry.

Does it mean, we have a one parameter family $\omega_t$, $t\in [a,b]$ of symplectic forms? If in that case why the word continuity though? Or does it mean we have a continuous map $f:[a,b]\to \Omega^2_{Symp}$, where $\Omega^2_{Symp}$ is the space of all symplectic forms on an ambient manifold. But in this case what is the topology of $\Omega^2_{Symp}$?

Can anyone please help clarify the definition for continuous family.

I have a strong sense that my second interpretation is the right one. But the thing is I don't know the topology of the space of differential forms $\Omega(M)$ on $M$. This might be a standard topology because I have seen in McDuff's book: Introduction to symplectic topology, assumes that $\Omega^2(M)$ is a topological space.

I have posted this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here.

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry.

Does it mean, we have a one parameter family $\omega_t$, $t\in [a,b]$ of symplectic forms? If in that case why the word continuity though?
Or does it mean we have a continuous map $f:[a,b]\to \Omega^2_\text{Symp}$, where $\Omega^2_\text{Symp}$ is the space of all symplectic forms on an ambient manifold? But in this case what is the topology of $\Omega^2_\text{Symp}$?

Can anyone please help clarify the definition for continuous family.

I have a strong sense that my second interpretation is the right one. But the thing is I don't know the topology of the space of differential forms $\Omega(M)$ on $M$. This might be a standard topology because I have seen in McDuff's book Introduction to symplectic topology, assumes that $\Omega^2(M)$ is a topological space.

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asked Sep 14, 2023 at 9:15

Uncool

What is the topology on the space of differential forms $\Omega^2(M)$?

I have posted this question on MSE one year ago here https://math.stackexchange.com/q/4368193/223051 , but till now I did not received an answer. Therefore I have decided to post it here.

I have difficulty in understanding the meaning of "A continuous family of symplectic forms". I have seen this in many papers on symplectic geometry.

Does it mean, we have a one parameter family $\omega_t$, $t\in [a,b]$ of symplectic forms? If in that case why the word continuity though? Or does it mean we have a continuous map $f:[a,b]\to \Omega^2_{Symp}$, where $\Omega^2_{Symp}$ is the space of all symplectic forms on an ambient manifold. But in this case what is the topology of $\Omega^2_{Symp}$?

Can anyone please help clarify the definition for continuous family.

I have a strong sense that my second interpretation is the right one. But the thing is I don't know the topology of the space of differential forms $\Omega(M)$ on $M$. This might be a standard topology because I have seen in McDuff's book: Introduction to symplectic topology, assumes that $\Omega^2(M)$ is a topological space.