Is it permissible to use the internet network that is automatically connected to the device, knowing that it belongs to a neighbor and covers the area?

If your use of your neighbor's network does not cause them additional financial burdens or a decrease in speed and efficiency of use, then it is permissible for you to do so, based on the general principle of mutual forgiveness among neighbors and the noble Hadith. It is more prudent to seek your neighbor's permission if possible. Nevertheless, it is better to obtain your own subscription to avoid problems and preserve the rights of the telecommunications company.