On Manifolds (2010-)

"A topological manifold of dimension k is a Hausdorff topological space M with a countable base such that for all x ∈ M, there exists an open neighborhood of x that is homeomorphic to an open set of Rk." (Stephen Lovett, "Differential Geometry of Manifolds", 2010)

"Roughly speaking, a manifold is a set whose points can be labeled by coordinates." (Gerardo F. Torres del Castillo, "Differentiable Manifolds: A Theoretical Physics Approach", 2010)

"One of the most important skills you will need to acquire in order to use manifold theory effectively is an ability to switch back and forth easily between invariant descriptions and their coordinate counterparts." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The fact that manifolds do not come with any predetermined choice of coordinates is both a blessing and a curse. The flexibility to choose coordinates more or less arbitrarily can be a big advantage in approaching problems in manifold theory, because the coordinates can often be chosen to simplify some aspect of the problem at hand. But we pay for this flexibility by being obliged to ensure that any objects we wish to define globally on a manifold are not dependent on a particular choice of coordinates. There are generally two ways of doing this: either by writing down a coordinate-dependent definition and then proving that the definition gives the same results in any coordinate chart, or by writing down a definition that is manifestly coordinate-independent (often called an invariant definition)." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"You can very generally think of a manifold as a space which is locally Euclidian - that means that if you look closely enough at one small part of a manifold then it basically looks like Rn for some n." (Jon P Fortney, "A Visual Introduction to Differential Forms and Calculus on Manifolds", 2018)