I suspect that the use of "there is" in this sentence is incorrect and that the ...

The following is the definition of a manifold from the mathematics textbook Analysis on Manifolds by James R. Munkres:

Definition. Let k>0. A k-manifold in R^n of class C^r is a subspace M of R^n having the following property: For each p ∈ M, there is an open set V of M containing p, a set U that is open in either R^k or H^k, and a continuous map α : U → V carrying U onto V in a one-to-one fashion, such that:
(1) α is of class C^r.
(2) α^(-1) : V → U is continuous.
(3) Dα(x) has rank k for each x ∈ U.

The author uses there is to introduce a plural noun phrase. Wouldn't there are instead be grammatically correct? Is the use of there is here acceptable English or simply a grammatical mistake?

The rough rule I follow is that the correct verb form ('is' or 'are') is decided by the plurality of the first item of the list (that follows the verb). This is sometimes called the 'proximity rule'. In your list, the first item is one singular item ('an open set V'). Thus 'is' is correct.

These are both OK:

There are two apples, an orange, and a peach in the bowl
There is a man, two girls, and a dog in the bus.