Question

Show by mean of a counter example that separability is not
hereditary property.
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Answer 1

Answer:

Separability and ccc are not hereditary.

Answer 2

Answer:

So every singleton set $\{ (x, -x) \} \subset L$ where is an open set. Any dense subset of must we such that for all , i.efor all . But the set of real numbers is uncountable, and the only dense subset of is itself! Therefore is not a separable topological space and so separability is not hereditary.

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