It's called the Western Wall, expert.<quoted text>
The Wailing Wall isn't a "basement." It is the only part left of the platform that the temple was built upon. In some places this "platform" was 9 stories high. Have you been there to see it? The platform was built as an extension of the hill (called mountains by the Jews). You're trying to nitpick technical details to prove your point without sufficient personal knowledge of the subject, which I happen to have.
Would you disbelieve your doctor if he said that a growth on your kidney wasn't a part of your kidney or would you deny his personal knowledge of the subject?
Save up some $$, go to Jerusalem, and see for yourself. If you do go, avoid the Gaza Strip (for obvious reasons) and Salah ed-Din St.
Some streets in extreme orthodox areas are blocked off on weekends.
Security is tight. Expect to be searched. Dress moderately to avoid suspicion. Ein Kerem is a neat little neighborhood on the west side with good food and some interesting art studios and galleries. Try it sometime. You might be surprised at what you learn.
And unless the pictures lie, it consists of stones still standing one atop another.
Why is that, do you suppose?