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If so, is the empty function from it to any other space considered a continuous function? I can't really convince myself either way.

user62748
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2 Answers2

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Yes, and Yes.

In all topological spaces the empty set and the space itself are open, so the topological space of the empty set which is the space itself is open. No problems there.

The empty function $f: \emptyset \rightarrow A$ maps the open set $\emptyset$ to ... $\emptyset \subset A$ Which is open. And the preimage of any open set of A is the open set $\emptyset$ so it is continuous.

fleablood
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  • I beg to differ: there exists one and unique map from the empty set to any other set, as maps are just certain subsets of a certain cartesian product of sets. The empty set fits perfectly well in this definition...and it is continuous by the topological definition of continuity. – DonAntonio Jun 17 '16 at 18:20
  • Yes, you were right and I was wrong. I was editing a corrected answer when you correctly commented. – fleablood Jun 17 '16 at 18:24
  • Thank you. Yet the map $;f;$ (as you denote it) is not onto $;A;$ , so it doesn't really matter that $;A;$ is open. In fact, the empty function $;f\in\emptyset\times A;$ is just $;f=\emptyset;$ , and it is onto $;\emptyset\subset A;$ , which of course is always open. – DonAntonio Jun 17 '16 at 18:29
  • Again, you are right. I hadn't really ever given much thought to the empty function. – fleablood Jun 17 '16 at 18:43
  • That empty set is an odd beast. The first time I came into more close touch with it was in linear algebra I, when I was officially announced the empty set is the basis for the trivial subspace $;{0};$ of any linear space. It took me into reading quite a bit on this. – DonAntonio Jun 17 '16 at 18:46
  • I don't have much problem with the empty set. What I find to be "odd beasts" are concepts (such as functions, or basis of vector spaces, or cross product) which are clearly "meant" to operate or be based upon values but which can be defined vaccuuously. A function can be thought of as a something that maps all points X or as a subset of X x Y but if there is nothing that can be mapped.... or nothing to "cross" with Y... that's weird. I had to pause on what $\emptyset x Y$ or what an ordered pair in $\emptyset x Y$ would be. (The empty set; and there are none.) – fleablood Jun 17 '16 at 18:54
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A topological space is a set $X$ and a collection of subsets $\tau$ of $X$ such that some axioms are satisfied. The axioms do not require the existence of any elements. (A group, for example is also a set with some axiomx, but here one of the axioms require the existence of an identity element.) So by definition the empty set is a topological space with only one topology on it.

Thomas
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