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How is this? - A manifold carries a structure of a topologic space?

Are you studying Kumaresan's book A Course in Differential Geometry and Lie Groups? (I think you mentioned somewhere that this is the book your course is going to use.)

@BAYMAX Isn't this directly in the definition of differentiable manifold?

Basically differentiable manifold is a topological spaces together with the atlases and charts.