Which one of the following is grammatically correct?
a) There is a theater, a cafe, and a shop near the park center.
b) There are a theater, a cafe, and a shop near the park center.
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This is a rule of thumb:
When talking about one thing only, then you have to use "there is".
When taling about uncountable things (mass noun) such as water, money, sky etc. normally is used to refer them as a singular. (there is water` there is money etc.)
For more detiles - read here:
Since your sentence detailes many things, then the correct option is:
There are: a theater, a cafe, and a shop near the park center.