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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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    $\begingroup$ You can use the topology of uniform convergence on compact sets with some number of derivatives, or all derivatives, or you can use Whitney's fine topology, which is quite different. (See Golubitsky and Guillemin; if I remember correctly, they have a nice description of the fine topology.) $\endgroup$
    – Ben McKay
    Commented Sep 14, 2023 at 9:21
  • $\begingroup$ @BenMcKay So, do you know which of these topologies McDuff used in her book? $\endgroup$
    – Uncool
    Commented Sep 14, 2023 at 9:52
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    $\begingroup$ On p. 324, McDuff and Salamon say that they are using the $C^{\infty}$ topology, for example, i.e. uniform convergence on compact sets, in any local coordinates, with any number of derivatives. At a brief glance, that is the only place I could find explicit mention of a topology on differential forms in their book. $\endgroup$
    – Ben McKay
    Commented Sep 14, 2023 at 12:18

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