(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Here are two assignments the solutions of which seem somewhat too easy, so I want to check them. :)

1) Is the topological space (Z, P(Z)), where Z is the set of integers, and P(Z) its partitive set, separable?

2) Is the topological space (N, U), where U = {0}U{Oi : i is in N}, Oi = {i, i+1, i+2, ...}, a first-countable space? ("0" denotes the empty set)

3. The attempt at a solution

1) It seems to me that the only set which is dense in Z is Z itself, since every element of P(Z) intersects it. Since, Z is countable, we conclude that (Z, P(Z)) is separable.

2) A space is first-countable if each point has a countable neighbourhood basis. Now, if i is some natural number, then the neighbourhoods of i are Oi, Oi-1 , ... , O1. This family is also a neighbourhood basis for i, since, if U is any neighbourhood of i, there exist one or more sets from the basis such that Oi is contained in U. Since the number of these sets is always countable, we conclude that (N, U) is first countable.

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# Homework Help: First countable space and separability question

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