TITLE: Superfluous definitions
QUESTION [17 upvotes]: It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative. 
For if a and b are elements of R, and writing + for the group operation then applying the distributive property one has
$$
\begin{align}
a+a+b+b&=a*(1+1)+b*(1+1)\\\\
&=(a+b)*(1+1)\\\\
&=(a+b)*1+(a+b)*1\\\\
&=a+b+a+b,
\end{align}
$$
whence $a+b=b+a$.
For educational purposes, are there more (not only algebraic) examples of such superfluous definitions?

REPLY [4 votes]: A complete lattice is a poset in which every subset has both an infimum and a supremum. Existence of the infimum for every subsets is already enough. 
P: Every element is a lower bound for the empty set, so the infimum is the largest element and a largest element exists, so every set has an upper bound. The infimum of all upper bounds is easily seen to be an upper bound- and a smallest one at that. 
Nonnegativity in the definitions of metrics and norms is redundant but often included.
P: $0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)$ for all x,y. The case of norms is similar.
A regular Hasudorff space is just a regular space that is $T_0$.
P: Take two points, one is not in the closure of the other. Separate that point and the closure of the other by open sets. This also separates the points.