# what is the topological dimension of an empty set?

• Aug 4th 2009, 07:26 AM
zzzhhh
what is the topological dimension of an empty set?
The following picture is extracted from Munkres' "Topology" P305.
http://i3.6.cn/cvbnm/dd/ef/6c/11096e...178584a807.jpg
I have problem when the space $\displaystyle X$ is the empty set $\displaystyle \emptyset$. In this case, the only two open covering is $\displaystyle \emptyset$ and $\displaystyle \{\emptyset\}$. For $\displaystyle \emptyset$, the only open refinement is $\displaystyle \emptyset$, which is also an open covering of $\displaystyle \emptyset$. For $\displaystyle \{\emptyset\}$, its only open refinements are $\displaystyle \emptyset$ and $\displaystyle \{\emptyset\}$, both of them are open coverings of $\displaystyle \emptyset$. Now the problem comes: what is the order of $\displaystyle \emptyset$ and $\displaystyle \{\emptyset\}$, respectively? or, what is the "some point" in the definition of order?
Here are some of my understandings:
(1)The only topology of $\displaystyle \emptyset$ is $\displaystyle \{\emptyset\}$, which is discrete. According to Ex 1. of this section(P315), it seems that dim $\displaystyle \emptyset$ should be 0?
(2)$\displaystyle \cup\emptyset=\emptyset$, so $\displaystyle \emptyset$ is a covering of $\displaystyle \emptyset$.
(3)The first-order logic formulation of "$\displaystyle \cal B$ is a refinement of $\displaystyle \cal A$" is: $\displaystyle \forall x(x\in{\cal B}\rightarrow{x}$ is a subset of some element of $\displaystyle \cal A)$. For $\displaystyle \cal B=\emptyset$, the left part of $\displaystyle \rightarrow$ would always be false, then this implication is vacuously true, and in turn the whole wff is true, so $\displaystyle \emptyset$ is a refinement of $\displaystyle \cal A$. Note: this logic is just like what we use to prove that the empty set is contained in any set.
(4)"open refinement" means that every element in the collection is open. Express it as a wff: $\displaystyle \forall x(x\in{\cal B}\rightarrow{x}$ is open). The same discussion as (3) shows that $\displaystyle \emptyset$ is an open refinement of $\displaystyle \cal A$. Additionally, $\displaystyle \emptyset$ is an open covering of $\displaystyle \emptyset$ for the same reason.
(5)Expressed in first-order logic wff, a natural number $\displaystyle m+1$ is the order of $\displaystyle \cal A$ if the following is true: $\displaystyle [\exists x(x\in{X}\wedge n(x)=m+1)]\wedge[\neg\exists x(x\in{X}{\wedge}n(x)>m+1)]$, where $\displaystyle n(x)$ is a function giving the number of elements of $\displaystyle \cal A$ that contains $\displaystyle x$, i.e., $\displaystyle n(x)=|\{A\in{\cal A}|x\in{A}\}|$. When $\displaystyle X=\emptyset, \exists x(x\in{X}\wedge ...)$ is always false, then no natural number can satisfy the definition of order. This is where my problem lies in.
Another book written by Engelking, "General Topology", says in P385 explicitly that dim $\displaystyle \emptyset$=-1. I was totally confused.
Please help me explain (in rigorous logic) what is the topological dimension of $\displaystyle \emptyset$, if it exists. Thanks!
• Aug 4th 2009, 12:16 PM
HallsofIvy
Having never read Munkres (completely), I got lost about half way through what you are writing but here's how I would look at it.

Generally speaking, the boundary of an n dimensional set has dimension n-1. That is the surface of a three dimensional object is two dimensional and the boundary of a two dimensional object is one dimensional (a curve). The boundary of a curve consists of its two endpoints: and points have dimension 0. The boundary of a finite set of points is the empty set: the dimension of the empty set is defined to be -1 in order to continue that pattern.
• Aug 5th 2009, 11:14 AM
zzzhhh
Thanks for your reply, your post looks similar to the last paragraph of this section, which mentions another inductive definition of dimension identical to the one in my post when the space is compact metrizable. But this is irrelevant to my question. The picture attached contains all related information. I need a rigorous definition and exposition, that is, in language of math -- mathematical logic, of the topological dimension of the empty space, or somewhat more essential, of the order of the empty collection when space X is empty too.
• Aug 5th 2009, 11:44 AM
HallsofIvy
I see a box with a red X in it but there is no "picture attached".
• Aug 6th 2009, 02:40 AM
zzzhhh
I'm very sorry for inconvenience. After shifting to another picture host, Can u see it now?
• Aug 7th 2009, 01:39 AM
Rebesques
Quote:

dim$\displaystyle \emptyset$=-1

I think this is a convention. The definition out of Munkres' book, as it stands, cannot be applied to the empty set, because of that existential quantifier:

Quote:

$\displaystyle [\exists x(x\in{X}\wedge n(x)=m+1)]\wedge\neg\exists x(x\in{X}{\wedge}n(x)>m+1)$