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First countable space and separability question

  1. Aug 30, 2010 #1

    radou

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    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.
     
  2. jcsd
  3. Aug 30, 2010 #2
    Your solutions to both parts are correct, but for both parts you are thinking too hard.

    For part 1: A countable space is always separable for the simple reason that the space itself is always a countable dense subset.

    For part 2: A space with only countably many distinct open sets is always first countable (in fact second countable) because every neighborhood basis at a point (or every basis for the topology) is contained in the (countable) topology.
     
  4. Aug 30, 2010 #3

    radou

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    Thanks, I have thought about these facts, they're simple and obvious, but I'm a beginner here so I'm cautious. :)
     
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