Notation

There is a trade off between rigor and clarity. If one gives too many details, the overall picture is obfuscated. At times, the notation in these posts will rely on filling in the proper variables with the right 'type' to make the formulae have any meaning. What this accomplishes in my notes is a form of multiple dispatch. If a symbol's type is readily available from context, then one does not need to expound on the types. For example, if I write,

\[ f : X \to Y \]

be a continuous function, then it is understood that the domain and codomain contains topological structure.

A sequence is a function \( a \colon \mathbb{N} \to \mathbb{R} \) denoted \( a_n \).

A sequence, \( a_{n} \), is convergent when:

\[ \exists a \in \mathbb{R} \forall \varepsilon > 0 \exists N \in \mathbb{N} \forall n \geq N . | a_n - a | < \varepsilon \]

We may denote this succinctly as,

\[ a_n \to a \]

Building from above, if we do write \( b_n \to b \), we will assume \( b_n \) is a sequence in some space obvious from context, and \( b \) is the limit of the sequence.

Given a predicate \( P \colon U \to 1 \) we may consider the collection of all objects satisfying the predicate, denoted by:

\[ \{ x \mid P(x) \} \]

As stated, this collection need not be a set. In most circumstances, we will use the axiom schema of comprehension which states that the following is a set given a set \( X \)

\[ \{ x \in X \mid P(x) \} \]

When obvious from context we will omit the superset \( X \).

\( U \) is an open subset of a space \( X \) is denoted \( U \opn X \)

Mathematical proofs are designed to be written in a way that is so precise that a computer may verify the proof. We humans are not computers, and perhaps it is unhelpful from a pedagogical point of view to humans as such. Here is a sentence that would not fly in modern writing:

\( f \colon s(f) \to t(f) \)

The reason for this definition not making sense is because we are defining \( f \) in terms of its own source and target. That is, the definition is self referential without any base case. The "true" way to write out such an object is,

\[ f \colon X \to Y \]

where \( X = s(f) \) and \( Y \supseteq t(f) \).

I will allow such writing on this website, if it is obvious how to reorder the statements to make coherent sense.