Why is the logistic regression decision boundary linear in X? The logistic regression model,

\begin{equation}
\operatorname{p}(X) = \frac{\operatorname{e}^{\beta_0 + \beta_1 X}}{1 + \operatorname{e}^{\beta_0 + \beta_1 X}}
\end{equation}

is said to create a decision boundary linear in $X$.

As far as I understand, only the logit is linear in $X$. Is this the reason the decision boundary is linear in X? If so, why? And if this is not the case, what is the reason for this phenomenon?

I am confused about this because the decision boundary can be expressed as:
\begin{equation}
\operatorname{p}(X) = a, a \in [0,\, 1]
\end{equation}
And $\operatorname{p}(X)$ is not linear in $X$.
The decision boundary $p(X)=a, a\in(0,1)$ is 

$$\beta_0+\beta_1X=\log\frac{a}{1-a}$$

which is linear in $X$.