Superfluous definitions 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?
 A: A unique factorization domain is typically defined as:
a domain $D$ such that each non-zero and non-invertible element $d$ can be factored as a product of irreducible elements.
And, for any factorizations $d=u_1 \dots u_m$ and $d = v_1 \dots v_n$ with irreducibles $u_i,v_i$ one has
that $m=n$ and there exists a permutation $\sigma$ of $\{1,\dots, n\}$ such that for all $i$ one has that $u_i$ and $v_{\sigma(i)}$ are associated.
Yet, this can be replaced by:
And, for any factorizations $d=u_1 \dots u_n$ and $d = v_1 \dots v_n$ with irreducibles $u_i,v_i$ one has that there exists a permutation $\sigma$ of $\{1,\dots, n\}$ such that for all $i$ one has that $u_i$ and $v_{\sigma(i)}$ are associated.
In other words, it is sufficient that all factorizations of an element with the same number of factors are essentially equal. The fact that there, then, cannot be any factorization with a different number of factors can be proved and thus not have to be included in the definition.
Side note: If one replaces 'domain' by 'commutative cancellative semigroup with identity' the weakened definition actually is different.
A: A Lie subgroup of a Lie group is usually defined as a subgroup which is also a submanifold.
But actually any closed subgroup of a Lie group is automatically a manifold, hence a Lie subgroup.
Similarly, any continuous group homomorphism between Lie groups is automatically smooth, i.e. a morphism in the category of Lie groups. 
NB Needless to say, the usual definitions (maybe due to Chevalley?) are there for very good reasons and  shouldn't be changed. I just find it interesting that formally weakening them doesn't change the concepts studied.
A: Common axioms for groups are associativity, existence of two-sided identity and existence of two-sided inverses. (Sometimes even uniqueness is required too.) However, it is enough to require associativity, existence of right identity and existence of right inverses.
If we mix directions and require right identity and left inverses, we get something not too far removed from a group (I'll leave it as an exercise...)
A: A $\sigma$ algebra of subsets of a set X is defined as a collection of subsets of X which is invariant by taking complements and denumerable unions. And which contains the empty set. 
But this last condition is (almost) superfluous. If there exists an element, say $A$, in the $\sigma$-algebra, then it must contain $(A\cup A^c)^c$, which is the empty set. Hence the sole purpose of requiring the empty set to be in the $\sigma$-algebra, is to deny the empty set the right to be a $\sigma$-algebra itself.
I don't really know why the empty set should not be a $\sigma$-algebra, and I don't see any result that would suddenly fail badly if we give the empty set this promotion.
EDIT: Bourbaki, Topology, ch5, section 6 no 3 (TG IX.60) does not require the empty set to be in the $\sigma$-algebra. So I think this is really superfluous.
A: 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.
A: A distributive lattice is usually defined as a lattice for which the operation $\vee$ distributes over $\wedge$, and dually $\wedge$ distributes over $\vee$. See https://mathworld.wolfram.com/DistributiveLattice.html and Link. However, either of these distributivity axioms implies the other, and Wikipedia in fact uses just one of them: https://en.wikipedia.org/wiki/Distributive_lattice.
A: A rectangle is defined as a quadrilateral having four right angles, but one could replace "four" in the definition with "three".
