Does staying in the single market post-Brexit mean freedom of movement?
When the UK leaves the EU, will staying in the single market also mean we have to keep open door immigration from the EU too?
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That is the EU's negotiation position indeed:
Leaders made it crystal clear today that access to the single market requires acceptance of all four freedoms, including the freedom of movement. There will be no single market "à la carte".
The 'four freedoms' in question are the freedom of movement of goods, people, services and capital over borders.
(Whether the EU will actually stand firm on this is anyone's guess. Methinks it will.)
Actually freedom of movement in the EU was part of the Treaty of Rome, but de-facto it was not established until the Schengen Agreement but United Kingdom and Ireland opted-out of the Schengen Agreement to retain control on their borders.
After March 2019 UK/EU may impose restrictions to "free of movement", as well as some other bilateral benefits (as common access to health services, etc).
The impact will not only be for those who want to work in the UK, but for Britons who want to work/retire in the EU (in Spain there are 770,000 British pensioners, against 110,000 Spanish residents in the UK)