So a manifold is a topological space with certain additional properties, so I may as well begin my outline with what a topological space is. To start us off, I'll define the

*power set*.

Let $$S$$ be a point set. The power set on $$S$$ is the set formed from all possible subsets of $$S$$. So if, for instance, $$S=\{a,b,c\}$$ then the power set

$$\mathcal{P}(S) = \{\{a\},\; \{b\},\; \{c\},\; \{a,b\},\; \{a,c\},\; \{b,c\},\;

\{a,b,c\},\;\emptyset\}$$.

You may note three things from this:

elements in $$\mathcal{P}(S)$$ are

*sets*;

$$S$$ and $$\emptyset$$ are

*always*subsets of $$S$$;

$$\{a\}$$ is called the

*singleton set*and is not the same as $$a$$.This is so, because, whereas $$a \in \{a\}$$ is always true, $$a \in a$$ is meaningless.

So a

**topology**$$T$$ on $$S$$ is a subset of $$\mathcal{P}(S), \;\;T\subseteq \mathcal{P}(S)$$ that satisfies the following;

finite intersection of elements (sets, recall) of $$T$$ are in $$T$$;

arbitrary union of elements in $$T$$ are in $$T$$;

$$\emptyset \in T$$;

$$S \in T$$.

The ensemble $$ S,T$$ is called a topological space. I'll give a couple of examples straight away,

If $$ T = \{\emptyset, S\}$$ then $$S,T$$ is said to have the trivial, concrete or indiscrete topology. In contrast, if $$T= \mathcal{P}(S)$$, then $$S,T $$ is said to have the discrete topologogy.

Elements in $$T$$ are called the

*open sets*in $$S,T$$. I'll get to closed sets in a bit.

Suppose for some arbitrary set $$X$$ I have $$Y$$ as a proper subset, that is $$Y \subset X$$. Then the set of elements in $$X $$ not in $$Y$$ is referred to as the

*complement*of $$Y$$ in $$X$$, and is written $$Y^c \subset X$$.

OK, so the closed sets in $$S,T$$ are simply those sets in the complement of $$T$$ in $$S,T$$. Note that the notion of openness and closedness is not exclusive: sets can also be both open and closed or neither. For example, the $$\emptyset$$ is open, as it is always in $$T$$, likewise $$S$$. But $$S$$ is the complement of $$\emptyset$$ and $$\emptyset$$ is the complement of $$S$$, so these sets are both open and closed.

Just in case this way of defining open and closed sets is making anyone feel queasy, let me say this. Elements in what's called the

**standard**topology on $$\mathbb{R}$$ are intervals of the form $$(a,b)$$. These are the open sets in $$\mathbb{R},T$$, which you can easily check satisfy the axioms above; we also recognize these as open intervals.

In particular, note that, by the axiom of arbitrary union, I must have $$(-\infty, \infty) \in T$$, which we recognize as the real line.

Often one doesn't really care what the precise topology on a set is, so It is usual to slightly abuse notation and write simply "let $$X$$ be a topological space".

So I guess that's enough definitions to be going on with. There may be more to follow, if you want, but don't worry; once we have an understanding of topological spaces, manifolds, in their simplest form, follow quite easily, I think.

Mmm. Looking back on this, I wonder if it isn't too dense to be comprehensible - I can promise you, though, that it is correct