The 1920 US census for Hennepin Co., Minnesota - shows Olina Cole - widowed female, 77 yrs old - born in Norway, immigrated to the US in 1867? - in her household also lives her son Ben Cole, 47 yrs old - born in Minnesota. (Oline or Olina would be a much more typical Norwegian given name.)
The 1910 US census makes her name look more like 'Alivia' Cole - lists her as a 67 yr old Norwegian born widow. This census lists her year of immigration as 1878. But her daughter in that household is 40 yrs old and was born in Minnesota - so one or the other must be in error.
I was unable to locate the widow in the 1900 US census.
It would be helpful for you to be able to narrow down the year of immigration - and when Anton and 'Olivia' were married? before they emigrated from Norway? were any children born in Norway? Have you found Anton ever listed with a patronymic surname? since it would be highly unlikely that he would have used a surname like 'Cole' in Norway. Did either of their obituaries list any specific location in Norway as their birthplaces? All of these clues may help you locate Anton and/or 'Oline' in the 1865 Norwegian census...