Unless otherwise specified, we will assume ZF axioms. Recall the following ZF axiom:
Axiom of Infinity:
We will call the set of natural numbers. That is,
We can then define and iterate as before, whence by applying the axiom of infinity again we can obtain and so on. All such sets generated by this process are called ordinals. This in turn gives us the canonical linear ordering of the ordinals where iff
Definition 1. An ordinal is a successor ordinal iff A limit ordinal is an ordinal which is not a successor ordinal.
Proposition 2. (Transfinite Induction) Let denote the class of all ordinals (in accordance to VBG notion of class) and be a class. If
- if is a limit ordinal and for all then
Proof. Since is a linear ordering on the ordinals, let be the least ordinal such that But then which is a contradiction. Hence
We can index ordinals with ordinals to generate the notion of a sequence of ordinals. We define an nondecreasing (nonincreasing) sequence of ordinals an ordered set of ordinals where iff
Definition 3. Let be an nondecreasing sequence of ordinals and be a limit ordinal and Then we define the limit of the sequence as
A dual definition can be defined for nonincreasing sequences, in which case the limits can be respectively distinguished as left and right limits. A sequence is continuous if for every limit ordinal in the indexing subclass we have
An example of a sequence which is not continuous may be one of the form where are both limit ordinals. So in this case
(since the sup is actually a max in this case).
Definition 4. (Ordinal Arithmetic) We define
It follows that addition and multiplication are both associative, but not commutative. In particular one can see that
Definition 5. For a set we define its cardinality, denoted as the unique ordinal with which the set has a bijection. The corresponding subclass of ordinals is called the class of cardinals.
Proposition 6. If then
Proof. For every define
Hence the mapping is a bijection between and
Hence in this context, Cantor’s theorem immediately follows:
Proposition 7. Let and Then
Proof. We have
Also if and are bijections, then we can define by and this is easily seen to be a bijection.
And if then we can define which is also seen to be a bijection.
It thus follows that addition and multiplication of cardinals are commutative (that is, ordinal operations are commutative on this subclass).
Above we said and But
for any finite ordinal with Knuth notation
having raisings of Consider the following convention for defining countable infinite ordinals:
They are all called countable infinite ordinals since for any one of them To clarify when we are talking about cardinal numbers versus ordinal numbers, we will use aleph notation: From Cantor’s theorem above, we know that if then That is, the ordinal corresponding to must be greater than all of the countable ordinals above, otherwise its cardinality would be This necessitates the notion of uncountable ordinals and corresponding uncountable cardinals. We use subscripts to characterize these: etc.
Definition 8. An infinite cardinal is a successor cardinal iff is a successor ordinal, and it is a limit cardinal iff is a limit ordinal.
Definition 9. Let be a limit ordinal. An increasing –sequence with a limit ordinal is cofinal in if And if is an ordinal, then we define its cofinality as
It is easy to verify that iff is a successor ordinal. Also and for any finite ordinal
Proof. Let Then (in particular it is the smallest such ). Now if then certainly Now since is cofinal in a subsequence of indices is cofinal in (where cofinality can be chosen to be ). So
But then is cofinal in That is,
Definition 11. An ordinal is regular if It is singular if it is not regular.
Corollary 12. If is a limit ordinal, then is a regular cardinal.
Theorem 13. If is an infinite cardinal, then
 Jech, Thomas. Set Theory. 3rd Edition. Springer Monographs in Mathematics. Springer-Verlag. 2000.
Recall for a function for we define the Fourier transform of by
and the inverse Fourier transform by
If then a.e..
Now if and with we say is the weak derivative of provided
The Sobolev space is then defined as the set of all functions such that for all These become Banach spaces under the norm
and Hilbert when whence we denote
For we can say that consists of such that
(using Fourier multipliers).
Bessel Potential Spaces:
We can generalize with and and define the Bessel potential space as all such that Note the fractional power of the Laplacian is defined by virtue of the fact that is dense in , so we say
These spaces satisfy our desire of being Banach (and Hilbert when ).
There is an alternative approach. Recall that the Holder space is defined as all functions such that
That is, it is the set of functions on which are and whose -th partial derivatives are bounded and Holder continuous of degree These spaces are Banach under the above norm. We can generalize the Sobolev spaces to incorporate similar properties. Let us define the Slobodeckij norm for with and by
The corresponding Sobolev-Slobodeckij space is defined as all functions such that
where This becomes a Banach space under the norm
 http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)
 Lieb, Elliot and Michael Loss. Analysis. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.
 Evans, Lawrence. Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.
Let be a monoid. Then is a pointed set with point Let be a map satisfying
Such a map is called a monoidal action, and we correspondingly call an -pointed set. Note that since it is a pointed morphism. We will also write for short.
Suppose acts on and that is a differential pointed set (so ). Now define the pointed set
with point defined by for all Then this is a differential pointed set with differential defined by
for all so We then define the cohomology of with coefficients in as
where of course is the equivalence class of maps under the relation iff
Let be an -pointed set. Consider the two sets
where iff or for some nonzero (Note the “or” gives us symmetry). belongs to both of these (as a class of elements in the latter case such that (since is never true for nonzero and )). The point (or trivial class) in is called the torsion subpointed set of with respect to
Proposition 1. Let and the pointed natural numbers be -pointed monoids. Then
Proof. Let and and define be defined by (since ). The map is injective since if then for all but this just implies As is a monoid with one generator, any morphism is determined by the image of But such a map induces a preimage under And we have
So is a morphism.
We don’t however have that without some way of moving natural numbers between components of (i.e. a tensor product). We can define a tensor product of two -pointed sets and which are monoids, denoted as with the usual bilinear relations. We will denote simple elements by It’s easy to verify that this is a monoid. We then have the result:
Proposition 2. Let the hypotheses of Proposition 1 hold, then
Proof. Define by Suppose Then
Also if then since action for all we have a preimage And lastly so it is a morphism.
If the functors and are left (right) exact functors in the category of pointed sets and has an injective (projective) resolution, then we can define the -pointed cohomology (homology) via the right (left) derived functors ().
We wish to address four of the Eilenberg-Steenrod axioms for this category. First note that for dimension we have where denotes the trivial pointed set.
If and are pointed sets, then we define the intuitive product pointed set as the pointed set If and are also differential sets with differentials and then we define the product differential component-wise: This is clearly a differential on which we in turn call the product differential set.
If we then define the relation on where iff then we have Hence we have
We can similarly define a formal sum of pointed spaces whose corresponding homology is just defined as
So we have two forms of additivity. We also have excision, for if is a pair of sets with a sub-differential set of and then
We lastly address homotopy and omit the long exact sequence as we have not developed a dimensional concept (although recall we constructed a short exact sequence under an assumption on ). Let be a map between paired sets. We say this is a paired set morphism if If we furthermore have that these are paired pointed sets and with and sub-pointed sets, a paired set morphism between them is a paired pointed morphism if we also have that If both are differential sets as well, we can define a paired differential morphism if we further have Let be a paired differential morphism between such sets. Then we would like for its restriction to be well-defined. Hopefully some combination of requirements like normality and being a paired differential morphism will work, but I have not yet convinced myself. If it worked, we would in turn just define two maps to be homotopic mod provided they agreed on
Let be a pointed set (i.e. there is a nullary operation where ).
Definition 1. A pointed set is a differential set if there is an endomorphism (i.e. ) such that for all We say two elements are homologous, denoted if
Proposition 2. The homologous relation is an equivalence relation.
Definition 3. Elements of the equivalence class are called cycles. Elements of the set are called boundaries. The homology of is defined to be the set
Note that all boundaries are cycles, and so can have a pointed set structure with point Now let the quotient set be pointed as and note that can be pointed since
Proposition 4. as pointed sets.
Proof. Since the equivalence class it follows that the map is injective. This map is also clearly surjective since is defined on which is partitioned via And of course we also have
Now suppose we have a sequence of pointed sets and a sequence of pointed set homomorphisms such that for all We will denote such a collection by and call it a pointed set complex. We can also define the th homology of the complex as
where we have the intuitive equivalence relation on each set with We will also call a pointed set complex exact if for all
Let be a sub-differential set of a differential set That is, We will call a normal sub-differential set if for we have
Definition 5. We define the relative homology of with respect to as the set
Theorem 6. If is a normal sub-differential set of then there are homomorphisms such that
is an exact complex, where denotes the trivial pointed set .
Proof. Let then and is not the boundary of any element in (excluding ). Since let us just define This map is fine provided is not the boundary of any element in But this is not possible since is normal, so the map is well-defined.
Now let Then and is not the boundary of another element in We want to send it to a cycle in which is not the boundary of an element in We already have that can’t be the boundary of something in since it’s not the boundary of anything in So we will define by
The image points are of course boundaries since the final map sends everything to It follows that for so we have a differential complex of homology pointed sets. To prevent confusion, we will denote the homology sets of this complex of homology sets by and Since the homology pointed sets will all have as their point, we will simply find the underlying sets. since is just the identity map, that is, For note that so we have that
Lastly we have and