Let $$D\in \mathbb {N}$$, $$q\in [2,\infty )$$ and $$(\mathbb {R}^D,|\cdot |,dx)$$ be the Euclidean space equipped with the D-dimensional Lebesgue measure. In this article, we establish the Fefferman–Stein decomposition of Triebel–Lizorkin spaces $$\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$$ with the help of the dual on function sets which have special topological structure. A function in Triebel–Lizorkin spaces $$\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$$ can be written as a specific combination of $$D+1$$ functions in $$\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D) \cap L^{\infty }(\mathbb {R}^D)$$. To get such a decomposition, first, some auxiliary function spaces $$\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$$ and $$\mathrm {WE}^{\infty ,\,q'}(\mathbb {R}^D)$$ are defined via wavelet expansions. It is shown that $$\begin{aligned} {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\subsetneqq L^{1}({\mathbb {R}}^D) \cup {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\subset \mathrm{WE}^{1,\,q}({\mathbb {R}}^D)\subset L^{1}({\mathbb {R}}^D) + {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}\end{aligned}$$and $$\mathrm {WE}^{\infty ,\,q'}(\mathbb {R}^D)$$ is strictly contained in $$\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)$$. Next, the Riesz transform characterization of Triebel–Lizorkin spaces $$\dot{F}^0_{1,\,q}(\mathbb {R}^D)$$ by the function set $$\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$$ is established. Then the dual of $$\mathrm {WE}^{1,\,q}({\mathbb {R}}^D)$$ is considered. As a consequence of the above results, a Riesz transform characterization of Triebel–Lizorkin spaces $$\dot{F}^0_{1,\,q}(\mathbb {R}^D)$$ by Banach space $$L^{1}({\mathbb {R}}^D) + {\dot{F}^0_{1,\,q}({\mathbb {R}}^D)}$$ is obtained. Although Fefferman–Stein type decompositions when $$D=1$$ was obtained by Lin et al. (Mich Math J 62:691–703, 2013), as was pointed out by Lin et al., the approach used in the case $$D=1$$ cannot be applied to the cases $$D\ge 2$$. In the latter cases, some new skills related to Riesz transforms are to be developed.