In the paper

GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2): 159-165,

Theorem 2.(b): $TP^n(\mathbb{R}P^1)$ is homeomorphic to $\mathbb{R}P^n$.

By page 161 line 2-line 3 of the above paper, the unordered configuration space of $k$ points on $\mathbb{R}P^1$, $C_k(\mathbb{R}P^1)=TP^k(\mathbb{R}P^1)\setminus TP^{k-2}(\mathbb{R}P^1)$.

My question: can we write $C_k(\mathbb{R}P^1)=TP^k(\mathbb{R}P^1)\setminus TP^{k-2}(\mathbb{R}P^1)=\mathbb{R}P^k\setminus \mathbb{R}P^{k-2}$?

(1). I note that $\mathbb{R}P^k\setminus \mathbb{R}P^{k-2}$ has only 3 cells of dimensions $k-2$, $k-1$, $k$.

On the other hand, by the paper

On the homology of configuration spaces, CF Bödigheimer, F Cohen, L Taylor, Topology, 1989,

(2). I obtain that $H^1(C_k(\mathbb{R}P^1);\mathbb{Z}/2)$ is non-trivial.

**Question.** (1), (2) are not compatible. Where is my mistake?