Members of the 80,000-strong Rail, Maritime and Transport (RMT) union announced a strike next week from the evening of October 14 to the evening of October 16.
The walkout coincides with strikes by health workers, civil servants and council workers across Britain in separate pay, jobs and spending cuts disputes.
The RMT said that insufficient progress had been made in talks over the closures, which the union says will risk hundreds of jobs and safety standards in the system known as the "Tube".
"RMT will not stand back and allow Government-driven austerity cuts to hollow out the Tube system and leave it as a dangerous shell."
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Cash was last month elected new general secretary of the RMT, replacing the powerful Bob Crow, famous for leading numerous strikes that shut down the underground system.
Cash promised there would be "no deviation" from Crow's strategy under his leadership.
The underground system management argue that few tickets are bought at booths due to machines and touch-payment systems, and say staff should be stationed on platforms instead.