# Initial Notes

These notes were also written in December 2020 just as I started to figure out how to use $\LaTeX$, and started to write things related to math. This was actually used in a Youtube video I made(now privatized however because it was pretty terrible)

# Defining the Topology

Our study of topological spaces shall begin with the definition of a
topology/topological space. The definition of the topology was under
debate during the early 20th century, due to the fact that we wanted a
definition that was both vague enough that all special case were
included, but also be limited enough so that it was useful and our
standard theorems still held true. For mathematicians, it is always a
big debate how general a definitions should be, as it can affect so much
in mathematics. Though eventually a common definition was agreed upon,
though there are different forms of it, which we may or may not cover in
this lecture or future lectures. The definition may seem quite vague,
but I believe the reader should be able to grasp it after some time and
practice. We define the **topology** $\mathcal{T}$ on a set X as the
collection of subsets of X such that: (1) Both $\varnothing$ and X are
within $\mathcal{T}$ (2) The union of the elements of any sub-collection
of $\mathcal{T}$ is also in $\mathcal{T}$ (3) The intersection the
elements of any finite sub-collection of $\mathcal{T}$ is also in
$\mathcal{T}$ Any set X for which a topology $\mathcal{T}$ has been
defined is called a **topological space**.

Well, properly a topological space is actually the ordered pair (X,$\mathcal{T}$), but $\mathcal{T}$ will be omitted and our space will just be notated by X, unless otherwise stated.

# Open and Closed Sets

In our study of topology, the open and closed set with be an integral
part of our discussion, so I believe it is best for both the reader and
I to define it as early as possible. If X is a topological space, we
call the set U of X an **open set** if U belongs to the collection
$\mathcal{T}$. Now, we can reach another way of defining the topological
space,

A topology $\mathcal{T}$ can be defined as the the set X with a collection of open sets such that both X and $\varnothing$ are open, and so that the arbitrary unions of open sets and finite intersections of open sets are both open.

We also have the closed set, which can be simply defined as a set who's
complement is an open set, or in mathematical terms, A subset C of a
topological space X is defined to be **closed** if:
$(X-C) \in \mathcal{T}$ Theorem A.2.1 Let X be a topological space, then
the following holds true: (1) X and $\varnothing$ are closed (2) Finite
unions of closed sets are closed (3) Arbitrary intersections of closed
sets are closed *proof*: (1) is easy to prove, as $X-X = \varnothing$,
which is open, implying X is closed. $X - \varnothing = X$, which is
open, therefore implying $\varnothing$ is also closed. (2) Given an
indexed family of closed sets $\{A_a\}_{a \in J}$ :

Since the sets $X - A_a$ are open, the right side of our equation represents the arbitrary union of open sets, and therefore one can clearly see that $\cap A_a$ is closed

consider the equation:

Here, we have $A_i$ which is open for i = 1,2,....,n. Our right hand side is the the finite intersection of open sets and is therefore open. Therefore, $\cap A_i$ is closed

Now we see the student's dilemma, how can a set be both open and closed? Well, this is just one of those things that are hard to think about and accept, but you must. The confusion will clear as you continue your journey through topology, so do not be too worried dear student.

We can discuss both the interior and closure of a set, the interior is
not so important, but as you will see later the closure will be of much
use to us: We define the **interior** of a subset C of the topological
space X to be the intersection of all open sets contained in C:

Here, Int means the interior of C. Now, we want to move on the the
closure of the subset C, which, as said earlier, is much more useful
than the interior, We define the **closure** of the subset C of the
topological space X is defined as the intersection of all closed sets
containing C:

Again, here Cl means closure of set C. There are actually more notations for the closure of the set, one of which is $\overline{A}$, and we shall use this for this for efficiency's sake.

# Defining and Using the Basis to Create a Topology

Sometimes when we want to define a topology, going through a classic
approach as we described is quite inefficient, or even impossible, so we
will use an idea called the basis to make our topology in a more
efficient manner. A **basis** $\mathcal{B}$ of a topological space X is
defined as the collection of subsets of X(called **basis elements**)
such that: (1) For each $x \in X$, there exist a basis element B
containing x (2) If $x \in B_1\cap B_2$, then there is a basis element
$B_3$ containing x such that $B_3 \subset B_1\cap B_2$ If $\mathcal{B}$
satisfies these two conditions, there is a **Topology $\mathcal{T}$
generated by $\mathcal{B}$** defined as such: As subset U of X is said
to be open(an element of $\mathcal{T}$) if for each $x \in U$ there
exists a basis $B\in \mathcal{B}$ such that $x\in B$ and $B \subset U$.
One must also remember that the basis elements themselves are elements
of $\mathcal{T}$.

Just to make sure, we will make sure that the collection generated by $\mathcal{B}$ is indeed a topology on X. We check our first condition, when U is an empty set, it vacuously satisfies the condition of openness. Similarly, for X there very clearly exists a basis element such that for all $x \in X$, $x \in B$ and $B \subset X$. Now, let's take an indexed family $\{U_a\}_{a\in J}$, and show that $U = \bigcup_{a \in J} U_a$ does indeed belong to $\mathcal{T}$.

Given a $x \in U$, there exists an index a so that $x \in U_a$. Since $U_a$ is open, there exist a basis element B such that $x \in B \subset U_a$, then $x \in B$ and $B \subset U$, showing us that the set U is open.

Finally, we want to show that for any two $U_1$ and $U_2$ in $\mathcal{T}$, $U_1 \cap U_2 \subset \mathcal{T}$. Given $x \in U_1 \cap U_2$, we want to assign a basis element $B_1$ such that $x \in B_1 \subset U_1$, and a basis element $B_2$ such that $x \in B_2 \subset U_2$. The second condition allows us to create a third basis so that $x\in B_3 \subset B_1 \cap B_2$. We can therefore we can say $B_3 \subset U_1 \cap U_2$, and since x is within $B_3$, we have proven that $U_1 \cap U_2 \in \mathcal{T}$ by definition.

Additionally, we shall seek to prove that any finite intersection of open sets in $\mathcal{T}$ is also in $\mathcal{T}$. The proof for n=1 is trivial; we should induct this, so we assume that this is true for n-1 and prove it is true for n. We start with:

By hypothesis, $U_1 \bigcap...\bigcap U_{n-1}$ belongs to $\mathcal{T}$;
By the result we just proved, the union of
$(U_1 \bigcap...\bigcap U_{n-1}$ and $U_n$ also belongs to
$\mathcal{T}$. The following lemma may make the result easier to
understand: Lemma A.3.1: For the basis $\mathcal{B}$ of the topology
$\mathcal{T}$ on the set X, $\mathcal{T}$ is defined as the collection
of all unions of elements of $\mathcal{B}$ *proof*: As we said earlier,
the collection of elements of $\mathcal{B}$ is also a part of
$\mathcal{T}$. One can clearly see that the union of the elements of
$\mathcal{B}$ is also in $\mathcal{T}$. Given $U \in \mathcal{T}$, we
can find an element $B_x \in B$ such that we have $x \in B_x \subset U$.
It can therefore be concluded that $U = \bigcup_{x \in U}B_x$, in other
words, U is equal to a union of the elements of $\mathcal{B}$.

Now, we want to be as clear as possible in mathematics, so I'd like to
remove any confusion regarding the use of the term basis. In linear
algebra, a basis is a way of describing a given vector as the linear
combination of of basis vectors. This is unique, unlike a basis in
topology, which is as one may be able to tell from the previous lemma,
not unique. So far, we have shown two different ways to create a
topology out of its basis, but now we will go in the opposite direction
and get the basis from the topology! Lemma A.3.2: Let X be a topological
space. Suppose we have a collection $\mathcal{C}$ of the open sets U of
X, so that for every U and $x \in U$ , there exist an element
$C \in \mathcal{C}$ such that $x \in C \subset U$. The collection
$\mathcal{C}$ is the basis for the topology $\mathcal{T}$ on X *proof*:
We must first show our collection $\mathcal{C}$ is a basis. The first
condition is very clearly satisfied, as for a given $x \in X$, since X
is an open set, there is by hypothesis an element C of $\mathcal{C}$,
such that $x \in C \subset X$. To check the second condition, let x
belong to $C_1 \bigcap C_2$, where of course $C_1$ and $C_2$ are
elements of $\mathcal{C}$. Since both $C_1$ and $C_2$ are open sets,
$C_1 \bigcap C_2$, and therefore, by hypothesis, there exists an element
$C_3$ such that $x \in C_3 \subset C_1 \bigcap C_2$

Now that we've achieved that, we want to prove that the basis $\mathcal{T}$' produced by $\mathcal{C}$ is the same as $\mathcal{T}$. One should note that if $U \in \mathcal{T}$ and $x\in U$, then by hypothesis there exists an element $C \in \mathcal{C}$, such that $x\in C \subset U$. It therefore follows that U belongs to the topology $\mathcal{T}$' by definition. Now, let's have some basis W belonging to the topology $\mathcal{T}$', then W is a union of elements of $\mathcal{C}$, as one can see using the lemma we just proved. Since each element of $\mathcal{C}$ also belongs to $\mathcal{T}$, and $\mathcal{T}$ is a topology, $W \in \mathcal{T}$.