Implication Details

Assumptions: finite powers, sequential limits

Conclusions: countable powers

Reason: We can write XNX^{\mathbb{N}} as the limit of the sequence ⋯→X3→X2→X→1\cdots \to X^3 \to X^2 \to X \to 1 with transition morphisms fn:Xn+1→Xnf_n : X^{n+1} \to X^n, (x1,…,xn+1)↦(x1,…,xn)(x_1,\dotsc,x_{n+1}) \mapsto (x_1,\dotsc,x_n), i.e., pifn=pip_i f_n = p_i for 1≤i≤n1 \leq i \leq n.