# Topological Spaces

Here is series of reading notes for I reading the book “Toplogy without tears”.
And I’m so sorry about my poor English skill which caused u reading difficulties.

## 1.1 Topology

Definitions 1.1.1

Let X be a non-empty set. A set τ of subsets of X is said to be a topology on X if

(i) $X$ and the empty set, $\emptyset$, belong to $\tau$ ,

(ii) the union of any (finite or infinite) number of sets in $\tau$ belongs to $\tau$, and

(iii) the intersection of any two sets in $\tau$ belongs to $\tau$. The pair $(X,\tau)$ is called a topological space.

Here is an example

Let $X$ = $\{a, b, c, d, e, f\}$ and
$\tau_1$ = $\{X, \emptyset, \{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}\}$.

Then $\tau_1$ is a topology on $X$ as it satisfies conditions (i), (ii) and (iii) of Definitions 1.1.1,and we can say that $(X,\tau_1)$ is a topological space.

Now it’s time for some special definitions.

Definitions 1.1.2

Let $X$ be any non-empty set and let $\tau$ be the collection of all subsets of $X$. Then $\tau$ is called the discrete topology on the set $X$. The topological space $(X,\tau)$ is called a discrete space.

We note that τ in Definitions 1.1.2 does satisfy the conditions of Definitions 1.1.1 and so is indeed a topology.

Observe that the set $X$ in Definitions 1.1.2 can be any non-empty set. So there is an infinite number of discrete spaces – one for each set $X$.

Definitions 1.1.3

Let $X$ be any non-empty set and $\tau$ = $\{X, \emptyset\}$. Then $\tau$ is called the indiscrete topology and $(X,\tau)$ is said to be an indiscrete space.

Once again we have to check that $\tau$ satisfies the conditions of 1.1.1 and so is indeed a topology.

We observe again that the set $X$ in Definitions 1.1.3 can be any non-empty set. So there is an infinite number of indiscrete spaces – one for each set $X$.

Definitions 1.1.4

$\tau_1$ consists of $\mathbb{N}$,$\emptyset$, and every set $\{1,2,…,n\}$, for n any positive integer. $\tau_1$ called the initial segment topology.

$\tau_2$ consists of $\mathbb{N}$,$\emptyset$, and every set $\{n,n+1,…\}$, for n any positive integer. $\tau_2$ is called the final segment topology.

ez to prove they satisfy the conditions of Definitions 1.1.1.

Proposition 1.1.5

If $(X,\tau)$ is a topological space such that, for every $x \in X$, the singleton set $\{x\}$ is in $\tau$ , then $\tau$ is the discrete topology.

Prove this proposition is not hard,so u can try to prove that.(ps: use mathematical induction)

## 1.2 Open Sets,Closed Sets and “Clopen” Sets

Rather than continually refer to “members of $\tau$ “, we find it more convenient to give such sets a name. We call them “open sets”.

Definition 1.2.1

Let $(X,\tau)$ be any topological space. Then the members of $\tau$ are said to be open sets.

so we have

Proposition 1.2.2

If $(X,\tau)$ is any topological space, then

(i) $X$ and $\emptyset$ are open sets,

(ii) the union of any (finite or infinite) number of open sets is an open set, and

(iii) the intersection of any finite number of open sets is an open set.

We shall also name the complements of open sets. They will be called “closed sets”.

Definition 1.2.3

Let $(X,\tau)$ be a topological space. A subset S of X is said to be a closed set in $(X,\tau)$ if its complement in X, namely $X \backslash S$, is open in $(X,\tau)$.

here is an example

Let $X$ = $\{a, b, c, d, e, f\}$ and
$\tau_1$ = $\{X, \emptyset, \{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}\}$.

the closed sets are $\emptyset$, $X$, $\{b,c,d,e,f\}$, $\{a,b,e,f\}$, $\{b,e,f\}$ and $\{a\}$.

If $(X,\tau)$ is a discrete space, then it is obvious that every subset of $X$ is a closed set. However in an indiscrete space, $(X, \tau)$, the only closed sets are $X$ and $\emptyset$.

Proposition 1.2.4

If $(X,\tau)$ is any topological space, then

(i) $\emptyset$ and $X$ are closed sets,

(ii) the intersection of any (finite or infinite) number of closed sets is a closed set and

(iii) the union of any finite number of closed sets is a closed set.

Warning. The names “open” and “closed” often lead newcomers to the world of topology into error. Despite the names, some open sets are also closed sets! Moreover, some sets are neither open sets nor closed sets!

Give an example

Let $X$ = $\{a, b, c, d, e, f\}$ and
$\tau_1$ = $\{X, \emptyset, \{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}\}$.

we see that

(i) the set $\{a\}$ is both open and closed;

(ii) the set $\{b, c\}$ is neither open nor closed;

(iii) the set $\{c, d\}$ is open but not closed;

(iv) the set $\{a,b,e,f\}$ is closed but not open.

In a discrete space every set is both open and closed, while in an indiscrete space
$(X,\tau)$, all subsets of $X$ except $X$ and $\emptyset$ are neither open nor closed.

To remind you that sets can be both open and closed we introduce the following definition.

Definition 1.2.5

A subset $S$ of a topological space $(X,\tau)$ is said to be clopen if it is both open and closed in $(X,\tau)$.

In every topological space $(X, \tau)$ both $X$ and $\emptyset$ are clopen.

In a discrete space all subsets of $X$ are clopen.

In an indiscrete space the only clopen subsets are $X$ and $\emptyset$.

## 1.3 The Finite-Closed Topology

It is usual to define a topology on a set by stating which sets are open.

However, sometimes it is more natural to describe the topology by saying which sets are closed.

Definition 1.3.1

Let $X$ be any non-empty set. A topology $\tau$ on $X$ is called the finite-closed topology or the cofinite topology if the closed subsets of $X$ are $X$ and all finite subsets of $X$;

that is, the open sets are $\emptyset$ and all subsets of $X$ which have finite complements.

example

If $\mathbb{N}$ is the set of all positive integers, then sets such as

are finite and hence closed in the finite-closed topology. Thus their complements

are open sets in the finite-closed topology.

Definitions 1.3.2 Let $f$ be a function from a set $X$ into a set $Y$ .

(i) The function $f$ is said to be one-to-one or injective if $f(x_1) = f(x_2)$ implies $x_1 = x_2$, for $x_1,x_2 \in X$;

(ii) The function $f$ is said to be onto or surjective if for each $y \in Y$ there exists an $x\in X$ such that $f(x)=y$;

(iii) The function $f$ is said to be bijective if it is both one-to-one and onto.

Definitions 1.3.3

Let $f$ be a function from a set $X$ into a set $Y$. The function $f$ is said to have an inverse if there exists a function g of $Y$ into $X$ such that $g(f(x))=x$, for all $x \in X$ and $f(g(y))=y$, for all $y \in Y$. The function $g$ is called an inverse function of $f$.

Proposition 1.3.4

Let $f$ be a function from a set $X$ into a set $Y$.

(i) The function $f$ has an inverse if and only if $f$ is bijective.

(ii) Let $g_1$ and $g_2$ be functions from $Y$ into $X$. If $g_1$ and $g_2$ are both inverse functions of $f$, then $g_1 = g_2$; that is, $g_1(y) = g_2(y)$, for all $y \in Y$ .

(iii) Let g be a function from $Y$ into $X$. Then $g$ is an inverse function of $f$ if and only if $f$ is an inverse function of $g$.

All functions map one point to one point. Indeed this is part of the definition of a function.

A one-to-one function is a function that maps different points to different points.

Definition 1.3.5

Let $f$ be a function from a set $X$ into a set $Y$. If S is any subset of $Y$ , then the set $f^{-1}(S)$ is defined by

The subset $f^{-1}(S)$ of $X$ is said to be the inverse image of $S$.

Note that an inverse function of $f : X \to Y$ exists if and only if $f$ is bijective. But the inverse image of any subset of $Y$ exists even if $f$ is neither one-to-one nor onto.

Definition 1.3.6

A topological space $(X,\tau)$ is said to be a T1-space if every singleton set $\{x\}$ is closed in $(X,\tau)$. Show that precisely two of the following nine topological spaces are T1-spaces.

Definition 1.3.7

A topological space $(X,\tau)$ is said to be a T0-space if for each pair of distinct points $a$, $b$ in $X$.