The <PERSON> algebra is defined for any vector space $V$ with quadratic form $q$ as $\operatorname{Cliff}(V;q) = T(V)/\langle x \otimes x + q(x) \rangle$ where $T(V) = \displaystyle \bigoplus_{n \ge 0} V^{\otimes n}$ is the tensor algebra on $V$. This makes sense in particular for the tangent space of any point on a manifold with the quadratic form coming from the metric, so to construct a <PERSON> bundle (a bundle over a pseudo-Riemannian manifold where each fiber is a <PERSON> algebra) we need only find a way to glue the fibers together to get the full vector bundle stucture.
The <PERSON> algebra on a vector space (with any choice of quadratic form) is canonically isomorphic as a vector space to the exterior algebra on that space (but not as an algebra). Extending this to vector bundles over a pseudo-Riemannian manifold, the <PERSON> bundle can be topologized by defining this graded fiberwise vector space isomorphism $\operatorname{Cliff}(TM;g) \rightarrow \Lambda(TM)$ and pulling back the topology on $\Lambda(TM)$ (which you already know as it's a subbundle of the tensor bundle), which makes this into an isomorphism of vector bundles (but with different algebra structures). That is to say, excluding multiplication, the <PERSON> and exterior bundles are the same, and in particular they're the same topologically. There are more natural ways to topologize the <PERSON> bundle, e.g. via the associated bundle construction, but this is a quick and dirty way to get it done.
There are topological issues nearby though which can have a big effect on physics. 364
If you want to construct an irreducible <PERSON> module (which is just a spinor bundle, something of clear importance for physics if e.g. you want to write down the Dirac equation), the base manifold (e.g. spacetime) needs to have a spin structure. Formally, this means that in signature $(m,n)$ there needs to be a lift of structure group from $\operatorname{SO}(m,n)$ to $\operatorname{Spin}(m,n)$ (its double cover). Such a lift exists if and only if the Stiefel–Whitney class $w_2 (TM) = 0$. Manifolds with such a lift are called "spin manifolds", and to even have global spinor fields and Dirac operators on them we need spacetime to be a spin manifold. 578