Are there smooth manifolds that do not admit a connection?

Question

This is more of a general conceptual question than may seem from the question.

I know that we can define a manifold $M$ as the set $M$ with a topology $O$ and a smooth atlas $A$: $(M,O,A)$. This manifold does not have a connection defined, and therefore we would call it a manifold "without connection".

However, when we say that $M$ doesn't have a connection, we really mean: We have not yet specified a connection. For example, if $M$ is the unit sphere in $R^2$, we have not yet specified $M$'s connection, but it nevertheless does have a very specific connection, namely the one that it inherits from Euclidean $R^3$ (I don't know if there are other connections possible on it).

So my question is: Are there also manifolds that don't have a connection in that sense? Meaning that we can not possibly define a connection on them?

Edit: reformulation of my question:

Are there also manifolds that don't admit a connection? Meaning that we can not possibly define a connection on them?

Answer 1

Every paracompact $n$-dimensional smooth manifold has a connection that can be defined locally and glued with a partition of unity. Consider a covering $(U_i)_{i\in I}$ of $M$ by an atlas such that $U_i$ is diffeomorphic to an open subset of $R^n$, you can define a connection on $U_i$ and glued them to obtain a connection on $M$ with a partition of the unity.