Subcomplexes of projective spaces

Question

Hatcher defines subcomplex of a CW complex to be a closed subspace $A$ subset $X$ that is a union of cells of $X$. $\mathbb{RP}^n$ has a cell complex structure $e^0 \cup e^1\cup \cdots \cup e^n$. So $\mathbb{RP}^k$ is a subcomplex. It is said that these are the only subcomplexes of $\mathbb{RP}^n$. Why is this true? Why is $ e^2 \cup e^4$ not a subcomplex?

Answer 1

In the cell structure of $\mathbb {RP}^n$ with one cell in each dimension, the attaching map $S^{k-1}\to X^{k-1}$ of the $k$-cell $e^k$ is surjective. This means we have $$\overline{e^k} = e^k \cup X^{k-1} = e^k \cup e^{k-1} \cup \dots \cup e^0 = X^k = \mathbb {RP}^k.$$

The union $e^2\cup e^4$ of open cells you suggest is not closed and its closure would be $$ \overline{e^2\cup e^4} = e^0\cup e^1\cup e^2\cup e^3\cup e^4 = X^4 = \mathbb {RP}^4. $$