I set up a website a couple of month ago. I didn't submit it or made any inbound link. I wanted to finish it first. How could Google have indexed part of it?
Google bots keep searching foe new content . if you have not added no index than you can't stop them for indexing .
not sure what platform you're using, but in wordpress there is an option to make your blog invisible to search engines
youd be suprised some times even registering it with certian companys they have an external link some where on site or like another sayd some blog software sends pings to spiders.
I suggest you use robots file to stop google and other SEs from indexing you site. When you finish the work on the site, you can then allow bots to visits your site.
You can request Google to index your site if it's not there in their index. Google search engine always search for new contents, new sites, even if you dont tell them.