Website opening with both http and https

Probably you want your site to open on https://

Checklist

1)
Dashboard > Settings > General

wordpress url and site url should be both https://

2) Install https://wordpress.org/plugins/better-search-replace/
run replace routline (remove dry run checkmark) on all tables
replacing
http://mysite
to
https://mysite

use your real site address here

3) Set 301 redirect in .htaccess

RewriteEngine On
RewriteCond %{SERVER_PORT} 80
RewriteRule ^(.*)$ https://www.yourdomain.com/$1 [R,L]

or other code, if you have some specific server settings.
If in trouble – ask your webhosting support

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Or just use some magic! Like Really Simple SSL plugin or any other similar.
It will go all the required steps with one click