Let \(p\) be a prime number and denote by \(\mathbb{Z}_p\) the \(p\)-adic integers. It is known that for profinite groups \(G\) there is a ring valued functor \(\mathbf{W}_G\); in the special case \(G = \mathbb{Z}_p$, $\mathbf{W}_{\mathbb{Z}_p}(k)\) is the set of \(p\)-typical Witt vectors over a field \(k\). A topology, that the authors call the initial vanishing topology, is defined on \(\mathbf{W}_{\mathbb{Z}_p^d}(k)\), where \(k\) is a field of characteristic \(p\). For \(d=1\), this topology agrees with the topology defined by the maximal ideal in \(\mathbf{W}_{\mathbb{Z}_p}(k)\). However, it is shown in this paper that for \(d \geq 2\), the initial vanishing topology and the topology defined by the maximal ideal in \(\mathbf{W}_{\mathbb{Z}_p^d} (k)\) do not agree. Another result proved in this paper is: There is a homomorphism from from \[\mathbf{W}_{\mathbb{Z}_p^2}(k) \rightarrow \mathcal{C}(\mathbb{P}^1(\mathbb{Q}_p), \mathbf{W}_{\mathbb{Z}_p}(k)), \] where \(\mathcal{C}(\mathbb{P}^1(\mathbb{Q}_p), \mathbf{W}_{\mathbb{Z}_p}(k))\) is the ring of continuous functions from \(\mathbb{P}^1(\mathbb{Q}_p)\) to the \(p\)-typical Witt vectors \(\mathbf{W}_{\mathbb{Z}_p}(k)\). Using this result the authors are able to show that the Krull dimension of \(\mathbf{W}_{\mathbb{Z}_p^d}(k)\) is infinite for \(d \geq 2\).