Let $(X,*)$ be a pointed topological space.

Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid x_i\neq x_j, i\neq j\}$.

Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.

Is there an inclusion $F(X,k)/S_k\to F(X,k+1)/S_{k+1}$ for each $k\geq 1$?

Note that $[x_1,\cdots,x_k]\mapsto [x_1,\cdots,x_k,*]$ is not well-defined.

This is true when $X=\mathbb{R}^n$. I want to try $X=S^n, \mathbb{R}P^n,\mathbb{C}P^n$...

Is there other way?