Will web sockets when used in all web browsers make ajax obsolete?
Cause if I could use web sockets to fetch data and update data in realtime, why would I need ajax? Even if I use ajax to just fetch data once when the application started I still might want to see if this data has changed after a while.
And will web sockets be possible in cross-domains or only to the same origin?
WebSockets will not make AJAX entirely obsolete and WebSockets can do cross-domain.
AJAX
AJAX mechanisms can be used with plain web servers. At its most basic level, AJAX is just a way for a web page to make an HTTP request. WebSockets is a much lower level protocol and requires a WebSockets server (either built into the webserver, standalone, or proxied from the webserver to a standalone server).
With WebSockets, the framing and payload is determined by the application. You could send HTML/XML/JSON back and forth between client and server, but you aren't forced to. AJAX is HTTP. WebSockets has a HTTP friendly handshake, but WebSockets is not HTTP. WebSockets is a bi-directional protocol that is closer to raw sockets (intentionally so) than it is to HTTP. The WebSockets payload data is UTF-8 encoded in the current version of the standard but this is likely to be changed/extended in future versions.
So there will probably always be a place for AJAX type requests even in a world where all clients support WebSockets natively. WebSockets is trying to solve situations where AJAX is not capable or marginally capable (because WebSockets its bi-directional and much lower overhead). But WebSockets does not replace everything AJAX is used for.
Cross-Domain
Yes, WebSockets supports cross-domain. The initial handshake to setup the connection communicates origin policy information. The wikipedia page shows an example of a typical handshake: http://en.wikipedia.org/wiki/WebSockets