The bookUnit 2 · Topology49 / 83

Topological spaces, bases, subspace/product/quotient

Why this is asked: Compare topologies (finer/coarser) and know how the product topology differs from the box topology on infinite products — that difference is examined directly.

Topologies, bases and the standard constructions

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Which properties survive subspaces, products and quotientsinteractive

A table you can reconstruct — and the operation that destroys Hausdorff.

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The trap here

“cl(A ∩ B) = cl(A) ∩ cl(B)” — false

$A = \mathbb{Q}, B = \mathbb{R}$ ∖ $\mathbb{Q}$ in $\mathbb{R}$

cl(A ∩ B) = cl(∅) = ∅ but cl A ∩ cl $B = \mathbb{R}$ .