General Topology - Intro and Continuous Functions

I hate topology. --- By Woziji Shuode

Topology

Definition 1. [Topology]

A topology on a set X is a collection of its subsets $\mathfrak{T}$, which has the following three properties:

1. $\emptyset \in \mathfrak{T}$ and $X \in \mathfrak{T}$;
2. For arbitrary subcollection $S$ of collection $\mathfrak{T}$, the union of $S$ is in $\mathfrak{T}$ too;
3. For any finite subcollection $S$ of $\mathfrak{T}$, the intersection of the elements of $S$ is in $\mathfrak{T}$ too.

When we are talking about some topological properties of any set $X$, we are using a certain topology $\mathfrak{T}$. In that case, we say "$X$ is a topological space with topology $\mathfrak{T}$". In the following content, we default topological space $X$ is with topology $\mathfrak{T}$.

Definition 2. [Open]

A subset of $X$ is said to be open if it is in the topology $\mathfrak{T}$.

Till now, the definition seems to be too abstract to understand, therefore let's give an example.

Example 3. Suppose we have a metric space $X = \mathbb{R}^2$, then $X$ has a topology $\mathfrak{T}$, which is the set of all its open subsets.(Here, we use the definition of open from Basic Topology) Then:

1. $\emptyset$ and $X$ are both open;
2. For arbitrary collection of open sets, their union is open;
3. For finite collection of open sets, their intersection is open.

Some topology is special:
Definition 4. [Discrete topology, Indiscrete / Trivial topology, Finite complement topology]

1. The collection of all subsets of $X$ is called discrete topology.
2. $\{X, \emptyset\}$ is called indiscrete topology or trivial topology.
3. let $\mathfrak{T}_f = \{U \subseteq X\: : \:U\:is\:finite\:or\:X\:-\:U\:is\:finite\}$. $\mathfrak{T}_f$ is a topology too.

Topologies are naturally collections of subsets, therefore there are clearly relationships of inclusion between them. For topologies, we have other words to describe them.

Definition 5. [Comparable, (Strictly) Finer, (Strictly) Coaser] For two topologies $\mathfrak{T}$ and $\mathfrak{T}'$:

1. If either $\mathfrak{T} \subset \mathfrak{T}'$ or $\mathfrak{T}' \subset \mathfrak{T}$, we called $\mathfrak{T}$ and $\mathfrak{T}'$ comparable;
2. If $\mathfrak{T} \subset \mathfrak{T}'$, we call $\mathfrak{T}$ is finer than $\mathfrak{T}'$.
3. If $\mathfrak{T}' \subset \mathfrak{T}$, we call $\mathfrak{T}$ is coarser than $\mathfrak{T}'$;
4. If $\mathfrak{T}$ and $\mathfrak{T}'$ are comparable while $\mathfrak{T} \neq \mathfrak{T}'$, they are either strictly finer or strictly coarser.

Basis

Despite of topology, we also have a concept basis, which is a bit "looser" than topology.

Definition 6. [Basis]

A basis of a set $X$ is a collection $\mathfrak{B}$ of subsets of $X$ such that:

1. $\forall x \in X$, $\exists B \in \mathfrak{B}$ such that $x \in B$;
2. $\forall x \in B_1 \cap B_2$, $\exists B_3 \in \mathfrak{B}$ such that $x \in B_3$ and $B_3 \subset B_1 \cap B_2$.

As its name, basis are the bases of topologies. If $\mathfrak{B}$ is the basis of topology $\mathfrak{T}$, we say $\mathfrak{T}$ is generated from $\mathfrak{B}$.

Lemma 7.

1. $\mathfrak{T} = \{U \subseteq X : \forall x \in U, \exists B \subset U\;s.t.\; x\in B\}$ is called "generated by $\mathfrak{B}$". In other words, it's the collection of all unions of elements of $\mathfrak{B}$.
   From this Lemma, we have the topology generated by a certain basis $\mathfrak{B}$ is unique.
2. Let $\mathfrak{B}$ be a collection of open sets of $X$ w.r.t. topology $\mathfrak{T}$, such that for any $U \in \mathfrak{T}$ and for any point $x$ in $U$ ($\forall x \in U$), $\exists B \in \mathfrak{B}$ such that $x \in B$. Then we say $\mathfrak{B}$ is a basis for the topology $\mathfrak{T}$ of $X$.
   Different from $\mathfrak{B} \rightarrow \mathfrak{T}$, a topology $\mathfrak{T}$ can have multiple bases. For example, $\mathfrak{T}$ is a basis for itself.

Definition 8. [Subbasis]

A subbasis $\mathfrak{S}$ of a topology $\mathfrak{T}$ is a collection of subsets of $\mathfrak{T}$, such that the union of all elements of $\mathfrak{S}$ equals to $X$. The topology $\mathfrak{T}'$ generated by $\mathfrak{S}$ is defined as the collection of all unions of finite intersections of elements of $\mathfrak{S}$.

The subbasis $\mathfrak{S}$ for $\mathfrak{T}$ can be seen as "the smallest set of subsets of $X$, such that after I completed its topology properties, it becomes $\mathfrak{T}$".

Subspace Topology

Definition 9. [Subspace Topology, Subspace Basis]

Let $X$ be a space with topology $\mathfrak{T}$, with basis $\mathfrak{B}$. Let $Y$ be a subset of $X$, then

1. $\mathfrak{T}' = \{U \cap Y: U \in \mathfrak{T}\}$ is a topology of $Y$;
2. $\mathfrak{B}' = \{B \cap Y: B \in \mathfrak{B}\}$ is a topology of $Y$;

Closed sets, Limit points

Definition 10. [Closed set] A set $A \subseteq X$ is said to be closed if $X - A$ (or $X \ A$) is open.

Lemma 11.

1. $\emptyset$ and $X$ are closed;
2. Arbitrary intersections of closed sets are closed;
3. Finite union of closed sets are closed;

Lemma 12.

1. Let $Y$ be a subspace of a topological space $X$, then $A$ is closed in $Y$ if and only if there is a set $Y \subseteq X$ closed in $X$ such that $A = Y \cap U$; (Important for problems!)
2. Let $Y$ be a closed subspace of $X$. A subset $A \subseteq Y$ is closed in $Y$ if and only if it is closed in $X$.

Definition 14. [Interior, Closure]

Suppose $X$ is a topological space and $A \subseteq X$, then we have:

1. The interior of $A$ is $Int(A) = \bigcup \{U \subseteq A\;|\;U\;is\;open\;in\;X\}$. That is, the interior of $A$ is the union of all open sets contained in $A$;
2. The closure of $A$ is $\bar{A} = \bigcap \{A \subseteq U\;|\;U\;is\;closed\;in\;X\}$. That is, the closure of $A$ is the intersection of all closed sets containing $A$;

Theorem 15. Let $A$ be a subset of a topological space $X$ with basis $\mathfrak{B}$, then:

1. $x \in \bar{A}$ iff every open set containing $x$ intersects $A$;
2. $x \in \bar{A}$ iff every $B \in \mathfrak{B}$ containing $x$ intersects $A$

Definition 16. [Neighborhood, Limit point] Let $A$ be a subset of topological space $X$, then

1. For $p \in X$, every open set $U$ containing $p$ is called a neighborhood of $p$;
2. Let $A$ be a subset of a topological space $X$ with basis $\mathfrak{B}$, then $p$ is a limit point of $A$ if every neighborhood $U$ of $p$ satisfies $(U \setminus \{p\}) \cap A \neq \emptyset$. The set of limit points of $A$ is written as $A'$/

Theorem 17. $\bar{A} = A \cup A'$.

Theorem 18. Let $A$ be a subset of topological space $X$, then T.T.F.E. :

1. $A$ is closed;
2. $\bar{A} = A$;
3. $A' \subseteq A$.

Sequential Limits and Hausdorff Space

Definition 19. [Convergence]

Let $\{x_j\}_{j=1}^{\infty}$ be a sequence of points of $X$, then we say $p$ converges to $p$ if for any neighborhood $U$ of $p$, there exists $n \in \mathbb{N}$ such that for any $j \geq n$, $x_j \in U$.

Definition 20. [Hausdorff Space]

A topological subspace $X$ is called Hausdorff space if for any $p_1, p_2 \in X$, $p_1$ has a neighborhood $U_1$, $p_2$ has a neighborhood $U_2$, such that $U_1 \cap U_2 = \emptyset$.

Theorem 21. If $X$ is a Hausdorff space, then T.T.F.E. :

1. Every finite subset $U \subseteq X$ is closed;
2. If $\{x_j\}_{j=1}^{\infty}$ converges, then its limit is unique;
3. For any subset $A \subseteq X$, $p$ is a limit point of $A$ iff every neighborhood contains infinitely many points of $A$.

Continuous Functions

Definition 22. [Continuous]

Let $X$, $Y$ be two topological spaces. Then a function $f: X \rightarrow Y$ is continuous iff for any open subset $U \subseteq Y$, $f^{-1}(U)$ is open in $X$.

Definition 23. [Homeomorphism]

Let $X$, $Y$ be two topological spaces. Then a bijection $f: X \rightarrow Y$ is homeomorphism if both $f$ and $f^{-1}$ is continuous.