Prove Of Basis Using Topological Sub-basis or Subspaces[MAT404 Question 1b]

I recently Posted a question to stackexchange concerning how to prove a basis of a topology using topological sub-basis[MAT404 Test question 1b]

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The proof is explained in details, i hope you enjoy it.
Click the link below to see proofs:
MAT404 QUESTION 1B SOLUTION