Recently, I started reading the book “Topology Without Tears” by Sidney Morris. It is such an exciting adventure. And as always, I write notes about what I learned.

## Topology

Definition. Let X be a non-empty set and $\mathcal{T}$ be a set of subsets of X, i.e. $\mathcal{T}\subseteq\mathcal{P}(X)$. We call $\mathcal{T}$ a topology on X if

1.   $X,\emptyset\in\mathcal{T}$
2. The union of any (finite or infinite, countable or uncountable) number of sets that are in $\mathcal{T}$ also belongs to $\mathcal{T}$
3. The intersection of any two sets that are in $\mathcal{T}$ also belongs to $\mathcal{T}$.

The pair $(X, \mathcal{T})$ is called a topological space.

When $\mathcal{T} = \mathcal{P}(X)$, it is called the discrete topology on X. Together they form the discrete topological space.

When $\mathcal{T} = \{\emptyset, X\}$, it is called the indiscrete topology on X. Together they form the indiscrete topological space.

Notice the article “the (in)discrete topo”, it means for a non-empty set X, there is exactly ONE such topo.

Theorem. If $(X,\mathcal{T})$ is a topological space (from now abbreviated as t.s.) such that for every element $x\in X$, the singleton set $\{x\}\in\mathcal{T}$, then $\mathcal{T}$ is a discrete topology (from now abbreviated as topo).

### Open sets, closed sets, clopen sets

If $(X,\mathcal{T})$ is a t.s., each element of $\mathcal{T}$ is called an open set. Meanwhile, we call a set $S\subseteq X$ a closed set if $X-S\in\mathcal{T}$, i.e. X-S is open.

Theorem. If $(X,\mathcal{T})$ is a t.s.,

1.  $\emptyset, X$ are closed sets
2. The intersection of any (finite or infinite) number of closed sets is a closed set
3. The union of any two closed sets is a closed set

If S is both an open and a closed set, we call it a clopen set.

• In a discrete t.s., any subset of X is clopen.
• In an indiscrete t.s., only $\emptyset$ and X are clopen.
• In any t.s., $\emptyset$ and X are clopen.

## The Finite-closed Topology

Definition. Let X be any non-empty set. A topo $\mathcal{T}$ on X is called the finite-closed topology, or cofinite topology on X if the closed subsets of X consist of X and all finite subsets of X.

which also means that the open sets are $\emptyset$ and subsets of X that have finite complements.

“consist of” is different from “include”. When way say “the closed subsets of X consist of X and all finite subsets of X”, it means no sets other than the specified sets are closed, only X and all finite subsets of X are closed.

Infinite sets are not necessarily open sets. If the complement of an infinite set is also an infinite set, it is definitely not an open set.

Let $\mathcal{T}$ be the cofinite topo on X. If X has at least 3 distinct clopen subsets, X is finite. ▶ Proof

We know that X and $\emptyset$ are always clopen. Let S be another clopen set. Since it is open, X-S is finite. Since it is closed, S is finite. Hence $X=S\cup (X-S)$ is finite.

## Some other topologies

### $T_1$-space (Fréchet space)

A topo space $(X,\mathcal{T})$ is called a $T_1$-space if every singleton set $\{x\}$ is closed. Another definition is that for every pair of distinct points a and b in X, there exists an open set containing a but not b, and there exists an open set containing b but not a. Examples:

• Any discrete space is a $T_1$-space, since $\mathcal{T}=\{S:S\subset X\}$ and $X-\{x\}\subset X$, which brings us to $X-\{x\}\in\mathcal{T}$ (open set), so $\{x\}$ is closed, $\forall x$.
• An infinite set with the finite-closed topo is a $T_1$-space, because every singleton set is finite, and in the finite-closed topo space every finite set is closed.

### $T_0$-space (Kolmogorov space)

A topo space $(X,\mathcal{T})$ is called a $T_0$-space if for each pair of distinct points a and b in X, there exists an open set containing a but not b, or there exists an open set containing b but not a. Examples:

• Any $T_1$-space is an $T_0$-space. For each pair of distinct points a and b, $\{a\}$ is open and does not contain b.
• With the same explanation, any discrete space is a $T_0$-space.

### Sierpiński space

It is a $T_1$-space on $X=\{0,1\}$ but not a $T_0$-space, i.e. $\mathcal{T}=\{\emptyset, \{0,1\}, \{0\}\}$ or $\mathcal{T}=\{\emptyset, \{0,1\}, \{1\}\}$

### Countable-closed space

Let X be an infinite set. The countable-closed topology is defined to have X and all countable subsets of X as its closed sets.

### Door space

A topo space $(X,\mathcal{T})$ is called a door space if every subset of X is open or closed.

## Some topologies on $\mathbb{R}$

• The discrete topology
• The indiscrete topology
• The finite-closed topology
•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval (-n, n) for n positive integer. ▶ Why?

We can safely say that $\mathcal{T}$ is countable. Let’s prove that any union of elements in $\mathcal{T}$ is also in $\mathcal{T}$.

$S = \bigcup_{n\in I} (-n, n),\ \mathrm{where}\ I\subseteq \mathbb{N}^+$

If I is finite, $S=(-m,m)$ where m is the maximum element in I, hence it is also in $\mathcal{T}$. Otherwise, I is infinite and equal to $\mathbb{N}^+$, hence $S=\mathbb{R}\in\mathcal{T}$.

It is easy to prove the intersection between two sets in $\mathcal{T}$ is also in $\mathcal{T}$.

•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval [-n, n] for n positive integer.
•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval $[n,\infty)$ for n positive integer.
•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval (-r, r) for r positive real.
•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval [-r, r] and interval (-r, r) for r positive real.
•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval [-n, n] and interval (-r, r) for n positive integer and r positive real.

### Not a topology on $\mathbb{R}$

•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval [-r, r] for r positive real. ▶ Why?

You can always find some infinite monotonic increasing sequence of real number that converges to a real value r, but never precisely reach r. The infinite union of sets derived from the sequence converges to $(-r,r)\notin\mathcal{T}$.

•  $\mathcal{T}$ consists of $\mathbb{R}, \emptyset$, and every interval (-q, q) for q positive rational. ▶ Why?

You can find some infinite monotonic increasing sequence of rational number that converges to an irrational number, e.g. partial sums of the Maclaurin series representation of $e$. The infinite union of sets derived from the sequence converges to $(-r,r)\notin\mathcal{T}$.