Is $\bigcup_{c \ge 1} \mathsf{DTime}(2^{cn})$ closed under polynomial reduction?

It's well known $$EXP$$ is closed under polynomial reduction. It means $$\bigcup_{c \ge 1} \mathsf{DTime}(2^{c^{n}})$$ is closed under polynomial reduction. But what about $$\bigcup_{c \ge 1} \mathsf{DTime}(2^{cn})$$? Is it also closed under polynomial reduction?

No, this would contradict the time hierarchy theorem. Note that any language in $$\mathsf{EXP}$$ is polynomial time reducible to some language in $$\mathsf{E}$$ by padding. Suppose $$L\in \mathsf{DTIME\big(2^{n^c}\big)}$$, then consider $$L'=\big\{x0^{|x|^c}\big|x \in L\big\}$$. Obviously $$L'\in \mathsf{E}$$ and $$L\le_p L'$$, thus if $$\mathsf{E}$$ is closed under polynomial time reductions we have $$L\in\mathsf{E}$$ and consequently $$\mathsf{E=EXP}$$.