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

- X ∈ τ and ∅ ∈ τ,
- the union of any (infinite or finite) number of sets in τ belongs to τ, and
- the intersection of any two sets in τ belongs to τ.

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

for example, Let N be the set of all natural numbers, let τ_{4} consist of N, ∅, and all *finite*
subsets of N. Then τ_{4} is *not* a topology on N as any infinite union of sets in τ is not belonged
to τ.

Let X be any non-empty set and let τ be the collection of all subsets of X, then τ is called the **discrete
topology** on X. The yopology space (X, τ) 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 τ must be the discrete topology.

Let X be any non-empty set and τ = {X, ∅}. Then τ is called the **indiscrete topology** and (X, τ) is
said to be an **indiscrete space**.

There are also infinite number of indiscrete spaces.

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

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

S = ∪_{x ∈ S} x.

Since S is in τ, then τ is the set of all subsets of X.

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

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

If (X, τ) is any topological space, then

- X and ∅ 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, τ) be a topological space. A subset S of X is said to be a **closed set** in (X, τ) if its
complement in X, namely X \ S, is Open in (X, τ).

If (X, τ) is any topological space, then

- ∅ 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, τ) is said to be **clopen** if it is both open and closed in (X,
τ).

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

Let X be any non-empty set. A topology τ 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(x_{1}) = f(x_{2}) implies x_{1}= x_{2}, for x_{1}, x_{2}∈ X; - The function f is said to be
**surjective**if for each y ∈ Y there exists an x ∈ 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 ∈ X and f(g(y)) = y, for all y ∈ 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 ∈ X and f(x) ∈ 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, τ) be a topological space and X a
non-empty set. Further, let f be a function from X into Y. Put τ_{1} = {f^{-1}(S) : S ∈ τ}. Then
τ_{1} is a topology on X.

A topological space (X,τ) is said to be a **T _{1}-space** if every singleton set {x} is closed
in (X, τ).

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 R is said to be open in the **euclidean topology on R** if it has the following
property:

- For each x ∈ S, there exist a, b in R, with a < b, such that x ∈ (a, b) ⊆ S.

We have following interesting collaries from this definition.

- Let r, s ∈ R with r < s. In the euclidean topology τ on R, (r, s) is an open set.
- (r, ∞) and (-∞, 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 R. - For each a and b in R with a < b, [a, b] is a closed set in the euclidean topology on R.
- R is a T
_{1}-space. - Z is a closed subset of R.
- Q is neither a closed subset nor an open subset of R.
- The only clopen subsets of R are only R and ∅.

A subset S of R is open if and only if it is a union of open intervals.

Let (X, τ) be a topological space. A collection B of open subsets of X is said to be a **basis** for
the topology τ 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.

τ itself is a basis for τ. We do have many bases for a topology. Generally, if B is a basis for τ, then let
B_{1} is a collection of subsets of X such that B ⊆ B_{1} ⊆ τ, then B_{1} is also a basis
for τ.

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 = ∪
_{b ∈ B}b and - for any b
_{1}, b_{2}∈ B, the set b_{1}∩ b_{2}is a union of members of B.

Generally, we can define a euclidean topology on R^{n} = {<x_{1}, x_{2}, …,
x_{n}> : x_{i} ∈ R, i = 1, 2, …, n} as B = {<x_{1}, x_{2}, …,
x_{n}> ∈ R^{n} : a_{i} < x_{i} < b_{i}, i = 1, 2, …, n} of
R^{n} with sides parallel to the axes.

Infact, we will often use open discs as a group of basis since it it the expantion of intervals on R.

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

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

Let B be a basis for a topology τ on a set X. Then a subset U of X is open if and only if for each x ∈ U there exists a b ∈ B such that x ∈ b ⊆ U.

Let B_{1} and B_{2} be bases for topologies τ_{1} and τ_{2}, respectively, on a
non-empty set X. Then τ_{1} = τ_{2} if and only if

- for each b ∈ B
_{1}and each x ∈ b, there exists a b' ∈ B_{2}such that x ∈ b' ⊆ b, and - for each b ∈ B
_{2}and each x ∈ b, there exists a b' ∈ B_{1}such that x ∈ b' ⊆ b.

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

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

Let A be a subset of a topological space (X, τ). A point x ∈ 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 ∈ X as its limit points.

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

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

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

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

For example, the closure of Q on R is R as every interval must contain one rational number.

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

So Q is a dense subset of R.

Let A be a subset of a topological space (X, τ). 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 ∈ τ and U ≠ ∅ then A ∩ U ≠ ∅).

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

Let A be a subset of a topological space (X, τ). A point x ∈ X is a limit point of A if and only if every neighbourhood of x contains a point of A different from x.

Let U be a subset of a topological space (X, τ). Then U ∈ τ if and only if for each x ∈ U there exists a V ∈ τ such that x ∈ V ⊆ U.