Let \(X\) be a non-empty set. A set \(\tau\) of subsets of \(X\) is said to be a **topology** on \(X\) if

- \(X \in \tau\) and \(\emptyset \in \tau\)
- the union of any (infinite or finite) number of sets in \(\tau\) belongs to \(\tau\), and
- the intersection of any two sets in \(\tau\) belongs to \(\tau\).

The pair \((X, \tau)\) is called a **topological space**.

for example, Let \(\mathbb{N}\) be the set of all natural numbers, let \(\tau\) consist of \(\mathbb{N}\), \(\emptyset\), and all finite subsets of \(\mathbb{N}\). Then \(\tau\) is not a topology on \(\mathbb{N}\) as any infinite union of sets in \(\tau\) is not belonged to \(\tau\).

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 \(X\). The topology space \((X, \tau)\) is called a **discrete space**.

There will be infinite number of discrete spaces.

If every infinite subset of an infinite subset is open or all infinite subsets are closed, then \(\tau\) must be the discrete topology.

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**.

There are also infinite number of indiscrete spaces.

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

We can see that if \(S\) be any subset of \(X\), then

\[S = \cup_{x \in S} x\]Since \(S\) is in \(\tau\), then \(\tau\) is the set of all subsets of \(X\).

Moreover, if all infinite subsets of an infinite set \(X\) is in topology \(\tau\), then \(\tau\) must be a discrete topology, the proof is related to open and close sets.

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

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

- \(X\) and \(\emptyset\) are open sets,
- the union of any (finite or infinite) number of open sets is an open set, and
- the intersection of any finite number of open sets is an open set.

We can find that infinite intersections of open sets may not be open.

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 \(X / S\), is open in \((X, \tau)\).

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

- \(\emptyset\) and \(X\) are closed sets,
- the intersection of any (finite or infinite) number of closed sets is a closed set and
- the union of any finite number of closed sets is a closed set.

A set can be either open or closed or open and closed or neither open nor closed.

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

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

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 \(X\).

While all finite sets are closed, not all infinite sets are open. If a topological space has at least 3 distinct clopen subsets, then \(X\) must be a finite set.

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

- The function \(f\) is said to be
**injective**if \(f(x1) = f(x2)\) implies \(x1 = x2\), for \(x1, x2 \in X\); - The function \(f\) is said to be
**surjective**if for each \(y \in Y\) there exists an \(x \in X\) such that \(f(x) = y\); - The function \(f\) is said to be
**bijective**if it is both injective and surjective.

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 \(f\).

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

- The function \(f\) has an inverse if and only if \(f\) is bijective.
- Inverse is unique.
- \(g\) is an inverse function of \(f\) if and only if \(f\) is an inverse function of \(g\).

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

\[f^{-1}(S) = \{x : x \in X \land f(x) \in S\}.\]The subset \(f^{-1}(S)\) of \(X\) is said to be the **inverse image** of \(S\).

There’s an interesting conclusion of topology and inverse image: Let \((Y, \tau)\) be a topological space and \(X\) a non-empty set. Further, let \(f\) be a function from \(X\) into \(Y\). Put \(\tau_{1} = \{f^{-1}(S) : S \in \tau\}\). Then \(\tau_{1}\) is a topology on \(X\).

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

A discrete space or an infinite set with the finite-set topology is a \(T_{1}\)-space.

A topological space 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.

A subset \(S\) of \(\mathbb{R}\) is said to be open in the E**uclidean topology on \(\mathbb{R}\)** if it has the following property:

- For each \(x \in S\), there exist \(a, b\) in \(\mathbb{R}\), with \(a < b\), such that \(x \in (a, b) \subseteq S\).

We have following interesting collaries from this definition.

- Let \(r, s \in \mathbb{R}\) with \(r < s\). In the Euclidean topology \(\tau\) on \(\mathbb{R}\), \((r, s)\) is an open set.
- \((r, \infty)\) and \((-\infty, r)\) are open sets for every real number \(r\).
- Not all open sets in Euclidean space are intervals.
- For each \(c\) and \(d\) in R with \(c < d\), \([c, d]\) is not an open set in \(\mathbb{R}\).
- For each \(a\) and \(b\) in \(\mathbb{R}\) with \(a < b\), \([a, b]\) is a closed set in the Euclidean topology on \(\mathbb{R}\).
- \(\mathbb{R}\) is a \(T_1\)-space.
- \(\mathbb{Z}\) is a closed subset of \(\mathbb{R}\).
- \(\mathbb{Q}\) is neither a closed subset nor an open subset of \(\mathbb{R}\).
- The only clopen subsets of \(\mathbb{R}\) are only \(\mathbb{R}\) and \(\emptyset\).

A subset \(S\) of \(\mathbb{R}\) is open if and only if it is a union of open intervals.

Let \((X, \tau)\) be a topological space. A collection \(B\) of open subsets of \(X\) is said to be a **basis** for the topology \(\tau\) if every open set is a union of members of \(B\).

\(B\) generates the whole topological space like basis in vector space in linear algebra.

\(\tau\) itself is a basis for \(\tau\). We do have many bases for a topology. Generally, if \(B\) is a basis for \(\tau\), then let \(B_1\) is a collection of subsets of \(X\) such that \(B \subseteq B_1 \subseteq \tau\), then \(B_1\) is also a basis for \(\tau\).

Let \(X\) be a non-empty set and let \(B\) be a collection of subsets of \(X\). Then \(B\) is a basis for a topology on \(X\) if and only if \(B\) has the following properties:

- \(X = \cup_{b \in B} b\) and
- for any \(b_1, b_2 \in B\), the set \(b_1 \cap b_2\) is a union of members of \(B\).

Generally, we can define a Euclidean topology on \(\mathbb{R}^{n} = \{<x_1, x_2, ..., x_n>\ : x_i \in \mathbb{R}, i = 1, 2, ..., n\}\) as \(B = \{<x_1, x_2, ..., x_n>\ \in \mathbb{R}^{n} : a_i < x_i < b_i, i = 1, 2, ..., n\}\) of \(\mathbb{R}^{n}\) with sides parallel to the axes.

In fact, we will often use open discs as a group of basis since it it the expansion of intervals on \(\mathbb{R}\).

A topological space \((X, \tau)\) is said to satisfy the **second axiom of countability** if there exists a basis \(B\) for \(\tau\), where \(B\) consists of only a countable number of sets.

Let \((X, \tau)\) be a topological space. A family \(B\) of open subsets of \(X\) is a basis for \(\tau\) if and only if for any point \(x\) belonging to any open set \(U\), there is \(b \in B\) such that \(x \in b \subseteq U\).

Let \(B\) be a basis for a topology \(\tau\) on a set \(X\). Then a subset \(U\) of \(X\) is open if and only if for each \(x \in U\) there exists a \(b \in B\) such that \(x \in b \subseteq U\).

Let \(B_1\) and \(B_2\) be bases for topologies \(\tau_1\) and \(\tau_2\), respectively, on a non-empty set \(X\). Then \(\tau_1 = \tau_2\) if and only if

- for each \(b \in B_{1}\) and each \(x \in b\), there exists a \(b' \in B_{2}\) such that \(x \in b' \subseteq b\), and
- for each \(b \in B_{2}\) and each \(x \in b\), there exists a \(b' \in B_{1}\) such that \(x \in b' \subseteq b\).

Let \((X, \tau)\) be a topological space. A non-empty collection \(S\) of open subsets of \(X\) is said to be a **subbasis** for \(\tau\) if the collection of all finite intersections of members of \(S\) forms a basis for \(\tau\).

It is better to say the elements of set \(X\) as point if \((X, \tau)\) is a topological space.

Let \(A\) be a subset of a topological space \((X, \tau)\). A point \(x \in X\) is said to be a **limit point** (or **accumulation point** or **cluster point**) **of \(A\)** if every open set \(U\), containing \(x\) contains a point of \(A\) different from \(x\).

It is easy to verify that discrete space has no limit point. In indiscrete space, a set with at least two point will have all \(x \in X\) as its limit points.

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is closed in \((X, \tau)\) if and only if \(A\) contains all of its limit points.

Let \(A\) be a subset of a topological space \((X, \tau)\) and \(A'\) the set of all limit points of \(A\). Then \(A \cup A'\) is a closed set.

Let \(A\) be a subset of a topological space \((X, \tau)\). Then the set \(A \cup A'\) consisting of \(A\) and all its limit points is called the **closure of \(A\)** and is denoted by \(\bar{A}\).

\(\bar{A}\) is the smallest closed set containing \(A\), this implies that \(\bar{A}\) is the intersection of all closed sets containing \(A\).

For example, the closure of \(\mathbb{Q}\) on \(\mathbb{R}\) is \(\mathbb{R}\) as every interval must contain one rational number.

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is said to be **dense** in \(X\) or **everywhere dense** in \(X\) if \(\bar{A} = X\).

So \(\mathbb{Q}\) is a dense subset of \(\mathbb{R}\).

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is dense in \(X\) if and only if every non-empty open subset of \(X\) intersects \(A\) non-trivially (that is, if \(U \in \tau\) and \(U \neq \emptyset\) then \(A \cap U \neq \emptyset\)).

Let \((X, \tau)\) be a topological space, \(N\) a subset of \(X\) and \(p\) a point in \(N\). Then \(N\) is said to be a **neighborhood** of the point if there exists an open set \(U\) such that \(p \in U \subseteq N\).

Let \(A\) be a subset of a topological space \((X, \tau)\). A point \(x \in X\) is a limit point of \(A\) if and only if every neighborhood of \(x\) contains a point of \(A\) different from \(x\).

Let \(U\) be a subset of a topological space \((X, \tau)\). Then \(U \in \tau\) if and only if for each \(x \in U\) there exists a \(V \in \tau\) such that \(x \in V \subseteq U\).

A topological space \((X, \tau)\) is said to be **separable** if it has a dense subset which is countable.

Let \(S\) be a subset of \(\mathbb{R}\) which is bounded above and let \(p\) be the supremum of \(S\). If \(S\) is a closed subset of \(\mathbb{R}\), then \(p \in S\).

Let \(T\) be a clopen subset of \(\mathbb{R}\). Then either \(T = \mathbb{R}\) or \(T = \emptyset\).

Let \((X, \tau)\) be a topological space. Then it is said to be **connected** if the only clopen subsets of \(X\) are \(X\) and \(\emptyset\).

\(\mathbb{R}\) is connected as above.

A topological space \((X, \tau)\) is disconnected if and only if there are non-empty open sets \(A\) and \(B\) such that \(A \cap B = \emptyset\) and \(A \cup B = X\).

Let \(Y\) be a non-empty subset of a topological space \((X, \tau)\). The collection \(\tau_Y = \{O \cap Y : O \in \tau\}\) of subsets of \(Y\) is a topology on \(Y\) called the **subspace topology** (or the **relative topology** or the **induced topology** or the **topology induced on \(Y\) by \(\tau\)**). The topological space \((Y, \tau_Y)\) is said to be a **subspace** of \((X, \tau)\).

Since the subspace will change the sets’ properties, we must make clear in what space or what topology when we are talking about properties like open and close.

Topology induced on \(\mathbb{Z}\) by Euclidean topology on \(\mathbb{R}\) is a discrete topology.

A topological space \((X, \tau)\) is said to be **Hausdorff** (or a **\(T_2\)-space**) if given any pair of distinct points \(a, b \in X\) there exists open sets \(U\) and \(V\) such that \(a \in U\), \(b \in V\), and \(U \cap V = \emptyset\).

every \(T_2\)-space is a \(T_1\)-space.

\(\mathbb{R}\) is a \(T_2\)-space, and any subspace of \(T_2\)-space is a \(T_2\)-space.

A topological space \((X, \tau)\) is said to be a **regular space** if for any closed subset \(A\) of \(X\) and any point \(x \in X / A\), there exist open sets \(U\) and \(V\) such that \(x \in U\), \(A \subseteq V\), and \(U \cap V = \emptyset\). If \((X, \tau)\) is regular and a \(T_1\)-space, then it is said to be a **\(T_3\)-space**.

Let \((X, \tau)\) and \((Y, \tau_1)\) be topological spaces. Then they are said to be **homeomorphic** if there exists a function \(f: X \to Y\) which has the following properties:

- \(f\) is a bijection
- for each \(U \in \tau_1\), \(f^{-1}(U) \in \tau\), and
- for each \(V \in \tau\), \(f(V) \in \tau_1\).

Further, the map \(f\) is said to be a **homeomorphism** between \((X, \tau)\) and \((Y, \tau_1)\). We write \((X, \tau) \equiv (Y, \tau_1)\).

Homeomorphism is a equivalence relation.

Every open interval \((a, b)\) is homeomorphic to \(\mathbb{R}\).

Any topological space homeomorphic to a connected space is connected.

So far we have known twelve properties preserved by homeomorphisms, they are:

- \(T_0\)-space, \(T_1\)-space, \(T_2\)-space, regular space and \(T_3\)-space
- the second axiom of countability
- separable space
- discrete and indiscrete space
- cofinite and cocountable space
- connectedness

If we want to prove two spaces are not homeomorphic, we just need to check whether one meets these properties and another not.

Also, homeomorphism must have the same cardinality on set and topology.

A subset of \(S\) of \(\mathbb{R}\) is said to be an **interval** if it has the following property: if \(x \in S\), \(z \in S\), and \(y \in \mathbb{R}\) are such that \(x \lt y \lt z\), then \(y \in S\).

A subspace \(S\) of \(\mathbb{R}\) is connected if and only if it is an interval.

If \(a, b, c,\) and \(d\) are real numbers with \(a \lt b\) and \(c \lt d\), then

- \((a, b) \not\equiv [c, d)\),
- \((a, b) \not\equiv [c, d]\), and
- \([a, b) \not\equiv[c, d]\).

Let \(f\) be a function mapping \(\mathbb{R}\) into itself. Then \(f\) is continuous if and only if for each \(a \in \mathbb{R}\) and each open set \(U\) containing \(f(a)\), there exists an open set \(V\) containing \(a\) such that \(f(V) \subseteq U\).

Let \(f\) be a mapping of a topological space \((X, \tau)\) into a topological space \((Y, \tau')\). Then the following two conditions are equivalent:

- for each \(U \in \tau'\), \(f^{-1}(U) \in \tau\);
- for each \(a \in X\) and each \(U \in \tau'\) with \(f(a) \in U\), there exists a \(V \in \tau\) such that \(a \in V\) and \(f(V) \subseteq U\).

Let \((X, \tau)\) and \((Y, \tau_1)\) be topological spaces and \(f\) a function from \(X\) into \(Y\). Then \(f : (X, \tau) \to (Y, \tau_1)\) is said to be a **continuous mapping** if for each \(U \in \tau_1\), \(f^{-1}(U) \in \tau\).

Continuous relation is transitive.

If \(f\) and its inverse are all continuous, then \(f\) is a homeomorphism.

Let \((X, \tau\)) and \((Y, \tau_1)\) be topological spaces, \(f : (X, \tau) \to (Y, \tau_1)\) a continuous mapping. \(A\) a subset of \(X\), and \(\tau_2\) the induced topology on \(A\). Further let \(g : (A, \tau_2) \to (Y, \tau_1)\) be the restriction of \(f\) to \(A\); that is, \(g(x) = f(x)\), for all \(x \in A\). Then \(g\) is continuous.

Let \((X, \tau)\) and \((Y, \tau1)\) be topological spaces and \(f : (X, \tau) \to (Y, \tau_1)\) surjective and continuous. If \((X, \tau)\) is connected, then \((Y, \tau_1)\) is connected.

A topological space \((X, \tau)\) is said to be **path-connected** (or **pathwise connected**) if for each pair of (distinct) points \(a\) and \(b\) of \(X\) there exists a continuous mapping \(f : [0,1] \to (X, \tau)\), such that \(f(0) = a\) and \(f(1) = b\). The mapping \(f\) is said to be a **path joining \(a\) to \(b\)**.

Every interval and \(\mathbb{R}^{n}\) is path connected.

Every path-connected space is connected.

Let \(f : [a,b] \to \mathbb{R}\) be continuous and let \(f(a) \neq f(b)\). Then for every number \(p\) between \(f(a)\) and \(f(b)\) there is a point \(c \in [a,b]\) such that \(f(c) = p\).

Let \(f\) be a continuous mapping of \([0,1]\) into \([0,1]\). Then there exists a \(z \in [0,1]\) such that \(f(z) = z\). \(z\) is a **fixed point**.

This is a special case of **Brouwer Fixed Point Theorem**.