I have a question on how Spivak defines the integration of a $k$-form on an oriented $k$-dimensional manifold. I've included the relevant section of Calculus on Manifolds below. I should mention that when Spivak writes "manifold" they are referring to an embedded submanifold in $\mathbb{R}^n$.
Let $\omega$ be a $k$-form on an oriented $k$-dimensional manifold $M$. If there is an orientation-preserving singular $k$-cube $c$ in $M$ such that $\omega =0 $ outside of $c([0,1]^k)$, we define $$\int\limits_M \omega = \int\limits_c \omega.$$ Theorem 5-4 shows $\int_M\omega$ does not depend on the choice of $c$. Suppose now that $\omega$ is an arbitrary $k$-form on $M$. There is an open cover $\mathcal{O}$ of $M$ such that for each $U\in \mathcal{O}$ there is an orientation-preserving singular $k$-cube $c$ with $U\subset c([0,1]^k)$. Let $\Phi$ be a partition of unity for $M$ subordinate to this cover. We define $$\int\limits_M \omega = \sum_{\varphi\in \Phi}\int\limits_M \varphi\cdot \omega$$ provided the sum converges as described in the discussion preceding Theorem 3-12.
My question is about the open cover $\mathcal{O}$. Spivak never defines what it means for a subset of a manifold $M$ to be open in $M$. In addition to this, partitions of unity subordinate to an open cover have only been constructed in Euclidean space.
If we assume instead that "open" above refers to open in the ambient space $\mathbb{R}^n$, then we encounter another problem. If $k<n$, then $M$ has measure $0$, hence we cannot have an open set $U\subset \mathbb{R}^n$ contained in $M$. However, Spivak insists that for each $U\in \mathcal{O}$ there is a singular cube $c$ with $U\subset c([0,1]^k)\subset M$.
Is it possible that Spivak meant $M\cap U\subset c([0,1])^k$ instead of $U\subset c([0,1]^k)$?