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A smooth manifold is a topological manifold equipped with an atlas whose transition maps are not just homeomorphisms, but also diffeomorphisms. This allows for the definition of differentiable functions and tangent vectors.

The Hausdorff condition ensures that points are 'separated', which means that each pair of distinct points can be separated by disjoint open sets. This is important for the uniqueness of limits and well-behaved topology.

A topological manifold is a topological space that is locally Euclidean, second-countable, and Hausdorff.

An orientable manifold is one that has a consistent choice of 'orientation' for its tangent spaces throughout. This means there exists an atlas for which all transition maps are orientation-preserving.

An $n$-dimensional topological manifold is a topological space that is locally homeomorphic to $\mathbb{R}^n$, meaning each point has a neighborhood that 'looks like' $n$-dimensional Euclidean space.

An atlas on a manifold is a collection of charts whose domains cover the manifold such that the transition maps between overlapping charts are homeomorphisms.

A space is locally Euclidean if every point has a neighborhood homeomorphic to an open subset of some Euclidean space $\mathbb{R}^n$.

Yes, a topological manifold can have a boundary. The boundary of a manifold is the set of points that have neighborhoods homeomorphic to open sets in the half-space $\mathbb{R}^n_+ = \{ (x_1, x_2, ..., x_n) \in \mathbb{R}^n \mid x_n \geq 0 \}$.

A chart is a pair consisting of an open subset of the manifold and a homeomorphism from this subset to an open subset of $\mathbb{R}^n$. Charts are used to describe the manifold's local structure through coordinates.