Subspaces of interval maps related to the topological entropy

For $a\in [0,+\mathrm{\infty })$, the function space $E_{\ge a}$ ($E_{>a}$; $E_{\leq a}$;$E_{<a}$)ofallcontinuousmapsfrom$[0,1]$ to itself whose topological entropies are larger than or equal to $a$ (larger than $a$; smaller than or equal to $a$; smaller than $a$) with the supremum metric is investigated. It is shown that the spaces $E_{\ge a}$ and $E_{>a}$ are homeomorphic to the Hilbert space $l_2$ and the spaces $E_{\le a}$ and $E_{<a}$ are contractible. Moreover, the subspaces of $E_{\le a}$ and $E_{<a}$ consisting of all piecewise monotone maps are homotopy dense in them, respectively.