Product of 2-sphere and real projective plane

Definition
This topological space is defined as the Cartesian product of the defining ingredient::2-sphere $$S^2$$ and the defining ingredient::real projective plane $$\R\mathbb{P}^2$$ equipped with the product topology. It is denoted $$S^2 \times \R\mathbb{P}^2$$ or as $$\R\mathbb{P}^2 \times S^2$$.

Homology groups
The homology groups with coefficients in integers are as follows:

$$H_p(S^2 \times \R\mathbb{P}^2; \mathbb{Z}) = \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0, 2 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p = 1, 3 \\ 0, & \qquad p > 3 \\\end{array}$$