For a topological manifold $X$ is it true that $X$ is a covering of $X\lor X$?

Question

Here is my question:

Let $X$ be a topological manifold. Is it true that $X$ is a covering of $X\lor X$ and $X\lor X\lor X$ and, so on.

I have a intuition, $\pi_1(X\lor X)=\pi_1(X)*\pi_1(X)$.

Answer 1

It is not true. Consider $X=\mathbb R$, then $X\vee X$ looks like a cross ($+$) and there can't be a local homeomorphism between the two, by considering the wegde point (where the lines cross).

Answer 2

This is not true. Let $X \lor X$ be formed by gluing two copies of $X$ at $p\in X$. If there exists such an covering $\pi :X\to X\lor X$, then there is an open neighborhood of $p$ in $X\lor X$ that is homeomorphic to $\mathbb R^n$. But this is not true. Indeed we showed that $X\lor X$ cannot be covered by any topological manifold.