MHB Sequences and their limits, convergence, supremum etc.

ertagon2
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Could someone check if my answers are right and help me with question 5?
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Are you sure about the third statement in 4?
Also, you probably have some idea about 5. Are you doubting between two possibilities?
 
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Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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