My key words end with this "> begin with <meta name="keywords" content="....................(Is this correct?)
The w3 validator does not pick it as an error.
I've checked against other websites they have "/>
Which is correct?
Either is fine. The "> ending is for HTML, the " /> ending is for XHTML, but the browser doesn't care which you use. Because you are using an HTML doctype, it's best to use "> in this particular case ... just to keep the validator happy (... not that the validator matters much).