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Let $(X,\mathcal{J})$ be a topological space and let $Y\subset X$,

Define the collection $\mathcal{J}'$ of subsets of $Y$ as $\mathcal{O}'\subset Y$ of the form $\mathcal{O}'=\mathcal{O}\cap Y$ where $\mathcal{O}\in\mathcal{J}$

Then $(Y,\mathcal{J}')$ is a topological space - Provided $Y\ne \emptyset$

I am happy with the proof but it doesn't use $Y\ne\emptyset$ anywhere, and I can't see what would break if $Y$ were the nullset, yes it'd be a pretty useless topology, but useless topoligies are still topologies.

I'm sure $(\emptyset,\{\emptyset\})$ is a topological space because:

  1. The entire set is a member of the topology
  2. The null set is a member of the topology
  3. Finite intersections are closed
  4. Unions are closed

So why is it saying "Provided $Y\ne\emptyset$"

Book: Mendelson Introduction To Topology, Dover, page 92, chapter 3 proposition 6.2.

This is probably a really simple question, but it bothers me that I cannot see why he writes this, so I must find out! I would of course call such a topology useless, but as I said, useless topologies are still topologies!

Krish
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Alec Teal
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    Some texts have a preference to exclude empty things as sub-things. Or perhaps the author excluded the empty space as a topological space by requiring every topological space to have at least one point. For a topological space the empty subset is certainly a subspace. – Ittay Weiss Dec 31 '14 at 20:07
  • Because this is not a good book. Use other book. The best one is Engelking's "General Topology" – user40276 Dec 31 '14 at 20:13
  • @user40276 i don't think so plus it is Out of Print *and isn't a dover book* – user153330 Dec 31 '14 at 20:15
  • @user153330 I cannot understand your objection. Excluding $\emptyset$ is very problematic from the categorial point of view. And, about the book, you can find a free copy on the internet, so there is no problem at all. – user40276 Dec 31 '14 at 20:18
  • @user40276 I hate the stupid "user" names, but that comment is not the point, I will read any book I like! – Alec Teal Dec 31 '14 at 20:19
  • @user153330 get a real username, also it is both in print and a dover book. – Alec Teal Dec 31 '14 at 20:19
  • @AlecTeal link please? (the only one i found so far is: http://www.amazon.com/General-Topology-Rysxard-Engelking/dp/0800202090 (out of print) and http://www.amazon.com/General-Topology-Sigma-Series-Mathematics/dp/3885380064 (costs more than 100 box)) – user153330 Dec 31 '14 at 20:22
  • @user153330 that book isn't relevant at all. – Alec Teal Dec 31 '14 at 20:27
  • @AlecTeal i was talking about that book because user40276 said: ***

    Because this is not a good book. Use other book. The best one is Engelking's "General Topology"***

     – user153330 Dec 31 '14 at 20:28
  • I can imagine it simplifies further developments like "let $Y$ be a topological subspace, and let $y\in Y$. You won't have to check for non-emptiness every time. – GPerez Jan 01 '15 at 01:18

1 Answers1

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The topological space $(\emptyset,\{\emptyset\})$ has many pleasant properties. For instance it is Hausdorff, normal, locally compact, connected, and so on.

Moreover it has also the property that, for every topological space $(X,\mathcal{T})$ there is a unique continuous map $(\emptyset,\{\emptyset\})\to(X,\mathcal{T})$, and this map is even an embedding.

This is not the same as singletons with the discrete topology, because continuous maps from them to a topological space are not unique.

Several texts define topological spaces to have nonempty support, but this is really not needed. However, if the text uses that definition, you have to stick with it, at least when following the proofs and doing the exercises.

It's just a matter of conventions. But, to justify the book's author, the empty set has too many pleasant properties for being really useful, apart from making some arguments straighter by not requiring the check that a space or subspace actually contains points.

egreg
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