Topological Spaces

$$\newcommand{\ts}{(X,\mathcal T)} \newcommand{\ms}{(X, d)} \newcommand{\minus}{\setminus} \newcommand{\clo}{\overline} \newcommand{\T}{\mathcal T} \newcommand{\nat}{\mathbb N} \newcommand{\B}{\mathcal B} \newcommand{\xn}{\{x_n\}_{n=1}^\infty} $$

1. Topological Spaces

Definition. Topology, Topological Space, and Open Sets.

A pair $\ts$ is called a topological space if $X$, as a set, is equipped with a collection $\T$ of subsets (called a topology) of $X$ that satisfies:

  1. $\emptyset\in\T$, $X\in\T$

  2. Arbitrary union is in $\T$. i.e.

    $$\{U_\alpha\}_{\alpha\in\Lambda}\subset\T\implies \bigcup_{\alpha\in\Lambda}U_{\alpha}\in\T $$
  3. Finite intersection is in $\T$, i.e.

    $$\{U_i\}_{i=1}^n\subset\T\implies \bigcap_{i=1}^nU_{i}\in\T $$

In this case, every $U$ in $\T$ is called an open set.

Definition. Metric topology

$\ms$ is a metric space, if $N_{\cdot}(\cdot)$ is the neighborhood introduced by $d$, let

$$\T:=\bigcup_{r>0}\bigcup_{x\in X}N_{r}(x) $$

then $\T$ is called a metric topology introduced by $d$.

Example.

  1. $\ms$ is a metric space, then $(X, \tau)$ is a topological space where $\tau$ is the metric topology of $X$ introduced by $d$.
  2. $(X, \{\emptyset, X\})$ is called the trivial space. (The reason that this space is called trivial will be shown in the part of Hausdorff Space)
  3. $(X, 2^X)$ is called the discrete space.

Definition. Subspace Topology and Subspace

$\ts$ is a topological space with $Y\subset X$. The the collection

$$\T_Y:=\{Y\cap U\mid U\in \T\} $$

is called a subspace topology of $\T$, and $(Y,\T_Y)$ is called a subspace of $\ts$ correspondingly.

When dealing with a space $X$ and a subspace $Y$ , one needs to be careful when one uses the term “open set”. Does one mean an element of the topology of $Y$ or an element of the topology of $X$? The following lemma is useful in this case.

Lemma. transitivity of openness
Let $Y$ be a subspace of $X$. If $U$ is open in $Y$ and $Y$ is open in $X$, then $U$ is open in $X$.
Pf. Since $U$ is open in $Y$, $U$ = $Y\cap V$ for some set $V$ open in $X$ . Since $Y$ and $V$ are both open in $X$ , so is $Y\cap V$.

Definition. Closedness

$\ts$ is a topological space, $S$ is called closed in $(X, \T)$ iff $X\setminus S\in \T$

Same in metric space, closedness is not contrary to openness.

Lemma. $\ts$ is a topological space, then the followings are true

  1. $\emptyset$ and $X$ are closed.
  2. arbitrary intersection of closed subsets are is closed.
  3. finite union of closed subsets is closed.

Lemma. Munkres Thm 17.2

$Y\subset X$ is a subspace of $X$, then $S$ is closed in $Y$ (with respect to the subspace topology) iff there exists a closed subset $C\subset X$ such that $S=C\cap Y$

One direction of this lemma is unexpectedly useful in proofs

Pf. If there exists a closed set $C$ in $X$ such that $S=C\cap Y$. Then $X\setminus C$ is open in $X$, so by definition of subspace topology, $(X\setminus C)\cap Y$ is open in $Y$. But $(X\setminus C)\cap Y=Y\setminus(C\cap Y)=Y\setminus S$ is open in $Y$, so $S$ is closed in $Y$.

Conversely, if $S$ is closed in $Y$, $Y\setminus S$ is open in $Y$, by definition of subspace topology, $\exists U\in\T_X$ such that $Y\setminus S=Y\cap U$ where $X\setminus U$ is closed in $X$, and $S=Y\setminus (Y\cap U)=Y\cap (Y\setminus U)=Y\cap (X\setminus U)$. So $X\setminus U$ is such a closed set.

Corollary Monkres Thm 17.3

$Y\subset X$ is closed. $S\subset Y$ is closed in $Y$ iff $S$ is closed in $X$.

Both directions instantly come from $S=Y\cap{\text{some closed set in X}}$ thus closed in $X$

Note that when we talk about closedness and openness in topological space, we always should specify what space is we comparing to relatively

Definition. Interior Point Set, Closure

$\T$ is a topology of $X$, the set of Interior Point of $S\subset X$ is defined as

$$S^\circ:=\bigcup\{U\in\T\mid U\subset S\} $$

With $S$ defined, the closure of $S$ is defined as

$$\clo S:=\bigcap\{E\subset X\mid S\subset X, \text{E is closed}\} $$

By definition we have

$$S^\circ\subset S\subset \clo S $$

Definition. Neighborhood

$\ts$ is a topological space, any element of $\{U\in\T\mid p\in U\}$ is called a neighborhood of $p\in X$.

Here we generalize the concept of neighborhood from metric space, that is, any open set containing the specific point is called a neighborhood of this point.

Definition. Limit point

$\ts$ is a topological space, a point $p$ is said to be a limit point of a set $A\subset X$, if every punctured (去心) neighborhood $U\setminus\{p\}$ of $p$ satisfies

$$(U\setminus\{p\})\cap A\neq \emptyset $$

The set of all limit points of $A$ is denoted by $A'$, called the derivative set of $A$.

Theorem. Localization Property of Closure

$\ts$ is a topological space, $A\subset \ts$, then $p\in \clo A$ iff every $U\in\T$ that contains $p$ must satisfy $U\cap A\neq \emptyset$

Pf. Instantly comes from the definition of closure.

Theorem. Closure is the union of a set itself and its derivative set

$\ts$ is a topological space, $X\supset A\neq \emptyset$, then $\clo A=A\cup A'$

Pf. If $p\in \clo A \minus A$, $\forall U\in\T$ that contains $p$ satisfies $U\cap A\neq \emptyset$, but $p\not \in A$, so $(U\minus \{p\})\cap A\neq \emptyset$, which means $p\in A'$

Theorem. $\ts$ is a topological space, $A\subset X$, TTFE

  1. $A$ is closed

  2. $\clo A = A$

  3. $A'\subset A$

Pf.

1 $\implies$ 2. By definition $\clo A$ is the smallest closed set containing $A$, so $A=\clo A$

2 $\implies$ 3. Shown by the above theorem

3 $\implies $ 1. By the above theorem $\clo A=A'\cup A$, so $A=\clo A$. But $\clo A$ by definition is closed.

2. Continuity

Definition. Continuous function

$\ts$, $(Y, \T')$ are two topological spaces. A function $f:X\to Y$ is called continuous on $X$ if

$$U\in\T'\implies f^{-1}(U)\in\T $$

Theorem (Munkres Thm 18.1). $f:X\to Y$ is a function, TTFE

  1. $f$ is continuous on $X$
  2. preimage of any closed set in $Y$ should be closed in $X$
  3. $\forall A\subset X$, $f(\overline A)\subset \overline {f(A)}$
  4. $\forall p\in X$, neighborhood $V$ of $f(p)$, there exists $U\subset X$ a neighborhood of $p$ such that $f(U)\subset V$.

Pf.

1 $\iff$2. Instantly comes from the fact that a set is closed iff it’s complement is open.

1 $\iff$4. Instantly comes from the fact that a neighborhood is simply an open set in a topological space. This is more generalized than that in metric space.

2 $\implies$3. Let $w\in f(\clo A)$, and let $V$ be a neighborhood of $w$. Taking $x\in \clo A$ such that $f(x)=w$, (applying 4.) there exists an neighborhood $U$ of $x$ such that $f(U)\subset V$. Then $U\cap A\neq \emptyset$ and $V\cap f(A)\neq \emptyset$,

Definition. homeomorphism

$X$, $Y$ are two topological spaces, a bijection $f:X\to Y$ is called a homeomorphism if $f$ and $f^{-1}:Y\to X$ are both continuous

Definition. Embedding

An injection $f:X\to Y$ with $Z=f(X)$ is called a topological embedding of $X$ in $Y$ if $f':X\to Z$ obtained by restricting the range of $f$ is a homeomorphism between $X$ and $Z$.

Theorem. (Munkres Thm 18.2) Rules for constructing continuous functions

Let $X, Y, Z$ be topological spaces,

  1. Constant functions are continuous, i.e.

    If $f :X\to Y$ maps all of $X$ into the single point $\{y_0\}$ of $Y$ , then $f$ is continuous.

  2. Inclusion maps are continuous, i.e.

    If $A$ is a subspace of $X$, the inclusion function $j : A \to X$ is continuous.

  3. Composition of continuous functions are continuous, i.e.

    If $f: X\to Y$ and $g: Y \to Z$ are continuous, then the map $g\circ f: X \to Z$ is continuous.

  4. Restriction of a continuous function to its subspace is continuous, i.e.

    If $f: X \to Y$ is continuous, and if $A$ is a subspace of $X$, then the restricted function $f|_A : A \to Y$ is continuous.

  5. Restriction of a continuous function to its subspace and expanding the range to its superspace is continuous

    Let $f:X\to Y$ be continuous. If $Z$ is a subspace of $Y$ containing the image set $f (X)$, then the function $g: X\to Z$ obtained by restricting the range of $f$ is continuous. If $Z$ is a space having $Y$ as a subspace, then the function $h: X\to Z$ obtained by expanding the range of $f$ is continuous.

  6. Local information of a continuous function

    The map $f: X\to Y$ is continuous if $X$ can be written as the union of open sets $U_\alpha$ such that $f |_{U_\alpha}$ is continuous for each $\alpha$.

Definition. Senquencing convergence

$\{x_n\}_{n=1}^\infty\subset X$ is said to be convergent to a point $x_0\in X$ if every neighborhood $V$ of $x_0$, there exists $N\in\nat$ s.t.

$$n>N\implies x_n\in V $$

We know that in any metric space, the sequential limit is unique. It’s natural to consider the uniqueness of sequential limits in an average topological space. In fact, this property doesn’s hold in a random topological space. But if we add some constraints to it, this property will hold.

Such spaces are called Hausdorff spaces

3. Hausdorff Spaces

Definition. Hausdorff Space

$\ts$ is called a Hausdorff Space if $\forall p,q\in X$, $p\neq q$ there exists open sets $U,V$ such that

  • $p\in U$
  • $q\in V$
  • $U\cap V\neq \emptyset$

Observation.

  • Any metric space is Hausdorff (proof)
  • Trivial space is not Hausdorff
  • Discrete space is Hausdorff

Theorem. Munkres Thm 17.8
Every finite point set in a Hausdorff space $X$ is closed.
Pf. Only need to show any singleton is closed (because finite union of closed sets is still closed) Let $\{x_0\}$ be a singleton in $X$. Let $x\in X$ be any arbitrary point with $x\neq x_0$, then $x$ and $x_0$ have disjoint neighborhoods $U$ and $V$, respectively. Since $U$ does not intersect $\{x_0\}$, the point $x$ cannot belong to the closure of the set $\{x_0\}$. As a result, the closure of the set $\{x_0\}$ is $\{x_0\}$ itself, so that it is closed.

Theorem. Hausdorff spaces are generalized “metric space”
$X$ is Hausdorff, then any converging sequence has a unique limit. Moreover, $p$ is a limit point $A$ iff any neighborhood of $p$ contains infinitely many points of $A$.
Pf. suppose $\xn$ converges to $x$, if $y\neq x$, it’s easy to prove that $\xn$ does not converge to $y$ by definition.

Theorem. Munkres Ex 2.17.12

Any subspace of a Hausdorff space is Hausdorff.

Instantly comes from the definition of subspace topology.