9.1. Topological Spaces

9.1.1. Introduction

Definition

Let \(X\) be a non-empty set. A set \(\mathcal{T}\) of subsets of \(X\) is said to be a topology on \(X\) if:

  1. \(X\) and the empty set, \(\phi\), belong to \(\mathcal{T}\),

  2. the union of any (finite or infinite) number of sets in \(\mathcal{T}\) belongs to \(\mathcal{T}\), and

  3. the intersection of any two sets in \(\mathcal{T}\) belongs to \(\mathcal{T}\).

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

Example

  • \(X = \{a,b,c,d,e,f\}\) and \(\mathcal{T} = \{X, \phi, \{a\}, \{c,d\}, \{a,c,d\}, \{b, c, d, e, f\}\}\).

Definition

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

Definition

Let \(X\) be any non-empty set and \(\mathcal{T} = \{X,\phi\}\). Then \(\mathcal{T}\) is called indiscrete topology and \((X, \mathcal{T})\) is said to be an indiscrete space.

Proposition

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

Remark

The intersection of any finite number of members of \(\mathcal{T}\) is a member of \(\mathcal{T}\).

Definition

Let \(\mathbb{N}\) be the set of all positive integers.

  • \(\mathcal{T}_1\) consists of \(\mathbb{N}, \phi\), and every set \(\{1,2,\dots,n\}\) for \(n\) any positive integer. This is called the initial segment topology.

  • \(\mathcal{T}_2\) consists of \(\mathbb{N}, \phi\), and every set \(\{n,n+1,\dots\}\) for \(n\) any positive integer. This is called the final segment topology.

9.1.2. Open Sets, Closed Sets, Clopen Sets

Definition

Let \((X, \mathcal{T})\) be any topological space. Then the members of \(\mathcal{T}\) are said to be open sets.

Proposition

If \((X,\mathcal{T})\) is any topological space, then

  1. \(X\) and \(\phi\) are open sets,

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

  3. the intersection of any finite number of open sets is an open set.

The intersection of infinite number of open sets need not be open.

Definition

Let \((X,\mathcal{T})\) be a topological space. A subset \(S\) of \(X\) is said to be a closed set in \((X,\mathcal{T})\) if its compliment in \(X\), namely \(X\setminus S\), is open in \((X,\mathcal{T})\).

Proposition

If \((X,\mathcal{T})\) is any topological space, then

  1. \(\phi\) and \(X\) are closed sets,

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

  3. the union of any finite number of closed sets is a closed set.

Example

  • \(X = \{a,b,c,d,e,f\}\) and \(\mathcal{T} = \{X, \phi, \{a\}, \{c,d\}, \{a,c,d\}, \{b, c, d, e, f\}\}\).

  • The closed sets are: \(\phi, X, \{b,c,d,e,f\}, \{a,b,e,f\}, \{b,e,f\}, \{a\}\)

  • \(\{a\}\) is both open and closed.

  • \(\{b,c\}\) is neither open nor closed.

  • \(\{c,d\}\) is open but not closed.

  • \(\{a,b,e,f\}\) is closed but not open.

Definition

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

  • In every topological space \((X,\mathcal{T})\), both \(X\) and \(\phi\) are always clopen.

  • In a discrete space all subsets of \(X\) are clopen.

  • In an indiscrete subspace the only clopen subsets are \(X\) and \(\phi\).

9.1.3. Finite-Closed Topology

Sometimes it is more natural to describe the topology by saying which sets are closed.

Definition

Let \(X\) be any non-empty set. A topology \(\mathcal{T}\) 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 \(X\); i.e., the open sets are \(\phi\) and all subsets of \(X\) which have finite complements.

Example

  • Let \(\mathcal{T}\) be a finite-closed topology over \(\mathbb{N}\).

  • \(\{1\}, \{5,7,6\}, \{2,4,6,8\}\) are finite and hence closed.

  • Their complements are open sets.

  • Set of even positive integers is not a closed set (its infinite), hence its complement, the set of odd positive integers, is not an open set.

  • While all finite sets are closed, not all infinite sets are open.

Remark

Let \(\mathcal{T}\) be the finite closed topology on a set \(X\). If \(X\) has at least 3 distinct clopen subsets, then \(X\) is a finite set.

9.1.4. Functions and topologies

Definitions

Let \(f\) be a function from a set \(X\) into a set \(Y\).

  1. The function \(f\) is said to be one-one or injective if \(f(x_1)=f(x_2) \implies x_1 = x_2, \forall x_1,x_2\in X\).

  2. The function \(f\) is said to be onto or surjective if for each \(y \in Y \quad\exists x \in X | f(x) = y\)

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

Definition

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 \(g(f(x))=x,\quad \forall x\in X\) and \(f(g(y)) = y, \quad\forall y \in Y\). The function \(g\) is called the inverse function of \(f\).

Proposition

Let \(f\) be a function from a set \(X\) into a set \(Y\).

  1. The function \(f\) has an inverse if and only if \(f\) is bijective.

  2. 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 \(g_1(y)=g_2(y) \forall y \in Y\).

  3. 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 \(g\).

Definition

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:

\[f^{-1}(S) = \{x : x \in X \text{ and } f(x) \in S\}\]

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

Remark

Let \((Y, \mathcal{T})\) be a topological space and \(X\) a non-empty set. Further, let \(f\) be a function from \(X\) into \(Y\). Put \(\mathcal{T}_1 = \{f^{-1}(S) : S \in \mathcal{T}\}\). Then \(\mathcal{T}_1\) is a topology on \(X\).

9.1.5. \(T_0\) and \(T_1\) spaces

Definition

A topological space \((X, \mathcal{T})\) is said to be a \(T_1\)-space if every singleton set \(\{x\}\) is closed in it.

Definition

A topological space \((X,\mathcal{T})\) is said to be a \(T_0\)-space if for each pair of distinct points \(a,b \in X\), either there exists an open set containing \(a\) and not \(b\) or there exists an open set containing \(b\) and not \(a\).

  • Every \(T_1\)-space is a \(T_0\) space.

Sierpinski space

  • Let \(X = \{0,1\}\)

  • \(\mathcal{T} = \{\phi, X, \{0\}\}\) is a \(T_0\) space.

  • \(\mathcal{T} = \{\phi, X, \{1\}\}\) is a \(T_0\) space.

  • Both of above are known as Sierpinski spaces.

  • \(\mathcal{T} = \{\phi, X, \{0\}, \{1\}\}\) is a \(T_0\) space as well as a \(T_1\) space.

9.1.6. Countable-closed topology

Definition

Let \(X\) be any infinite set. The countable closed topology is defined to be the topology having as its closed sets \(X\) and all countable subsets of \(X\).

9.1.7. Unions and intersections

Let \(\mathcal{T}_1\) and \(\mathcal{T}_2\) be two topologies on \(X\).

  • \(\mathcal{T}_3 = \mathcal{T}_1 \cup \mathcal{T}_2\) need not be a topology on \(X\).

  • \(\mathcal{T}_4 = \mathcal{T}_1 \cap \mathcal{T}_2\) is a topology on \(X\).

  • If \((X, \mathcal{T}_1)\) and \((X, \mathcal{T}_2)\) are \(T_1\)-spaces then \((X, \mathcal{T}_4)\) is also a \(T_1\) space.

  • If \((X, \mathcal{T}_1)\) and \((X, \mathcal{T}_2)\) are \(T_0\)-spaces then \((X, \mathcal{T}_4)\) need not be a \(T_0\) space.

  • Intersection of any finite number of topologies on \(X\) is a topology on \(X\).

  • If for each \(i\in I\) for some index set \(I\), each \(\mathcal{T}_i\) is a topology on the set \(X\), then \(\mathcal{T} = \cap_{i \in I} \mathcal{T}_i\) is a topology on \(X\).