The mapping

$$\Phi:\varphi\in PSH(X,\omega)\mapsto\omega_{\varphi}:=\omega+dd^{c}\varphi\in{\mathcal{T}}_{[\omega]}(X)$$

is a continuous affine mapping whose kernel consists of constants mappings: indeed $\omega_{\varphi}=\omega_{\psi}$ implies that $\varphi-\psi$ is pluriharmonic hence constant by the maximum principle. Moreover $\Phi$ is surjective: if $\omega^{\prime}\geq 0$ is cohomologous to $\omega$ then $\omega^{\prime}=\omega+dd^{c}\varphi$ for some $\varphi\in L^{1}(X,\mathbb{R})$ -this is the celebrated $dd^{c}$-lemma on Kähler manifolds (see e.g. lemma 8.6, chapter VI in [15]). Thus $\varphi$ coincides almost everywhere with a function of $PSH(X,\omega)$ and $\omega^{\prime}=\Phi(\varphi)$. ∎