The SEC is currently open and operational despite the recent federal government shutdown. So far the federal government shutdown has not slowed the proposed Twitter IPO, as their Form S-1 registration statement was posted to the SEC website on October 3.
The SEC has released its operational plan in the event of a future SEC shutdown, which can be found here. If there is a lapse in appropriations resulting in an SEC shutdown, the SEC will continue only certain functions, which are discussed both in the SEC release referenced above and detailed by my colleagues here. Any changes in the SEC’s operational status will be announced on the SEC website.