i can't sign in to the ap store

One may have picked up a piece of Malware or Adware blocking the connection to the Ap Store.

Suggest downloading from a Respected ASC Contributor the application Malwarebytes for Mac. It is free or paid for added features. Run the Application and it should remove the malware / adware. Once done, restart computer and test.

There are no known Windows-like Viruses in the wild that self replicate and affect macOS, because of the underling UNIX  Foundation and Permission Limitation. 

There are Malware-viruses and Adware-viruses that does affect macOS and are often times downloaded as part of an Application from Third Party UnTrusted Site and get installed along with the Application.

lbwieland Said:

"i can't sign in to the ap store: I'm trying to open the ap store from my macbook air and I keep getting the message that I can't connect. Can anyone help?"

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Use Ethernet:

Connect using an Ethernet cable, rather than using WiFi. Ethernet is hardwired, meaning it is a more reliable and trustworthy source of connectivity.

Are you trying to get Big Sur?

macOS Big Sur - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

35.5GB available storage on macOS Sierra or later*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Big Sur:

MacBook (2015 or later)

MacBook Air (2013 or later)

MacBook Pro (Late 2013 or later)

Mac mini (2014 or later)

iMac (2014 or later)

iMac Pro (2017 or later)

Mac Pro (2013 or later)

Learn about compatibility and how to identify your specific Mac model

Don't use FireFox or WaterFox for this link. it'll mess it up...

https://apps.apple.com/us/app/macos-big-sur/id1526878132?ls=1&mt=12

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/