## Definition of Topology

Let $$X$$ be a non empty set. A collection $$\tau $$ of subsets of $$X$$ is said to be a… Click here to read more

From basic to higher mathematics

Let $$X$$ be a non empty set. A collection $$\tau $$ of subsets of $$X$$ is said to be a… Click here to read more

If $${\tau _1}$$ and $${\tau _2}$$ are two topologies defined on the non empty set X such that $${\tau _1}… Click here to read more

Indiscrete Topology The collection of the non empty set and the set X itself is always a topology on X,… Click here to read more

The intersection of any two topologies on a non empty set is always topology on that set, while the union… Click here to read more

Usual Topology on $$\mathbb{R}$$ A collection of subsets of $$\mathbb{R}$$ which can be can be expressed as a union of… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then a member of $$\tau $$ is said to be an… Click here to read more

Let $$X$$ be a non empty set, and then the collection of subsets of $$X$$ whose compliments are finite along… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then a subset of X whose complement is a member of… Click here to read more

We shall describe a method of constructing new topologies from the given ones. If $$\left( {X,\tau } \right)$$ is a… Click here to read more

Let $$X$$ be a topological space with topology $$\tau $$, and $$A$$ be a subset of $$X$$. A point $$x… Click here to read more

Let $$A$$ be a subset of a topological space $$X$$, then a point $$x \in A$$ is said to be… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space and $$A$$ be a subset of $$X$$, then the closure of… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space. A subset $$N$$ of $$X$$ containing $$x \in X$$ is said… Click here to read more

Let $$\left( {X,\tau } \right)$$ be the topological space and $$A \subseteq X$$, then a point $$x \in A$$ is… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space and $$A$$ be a subset of $$X$$, then a point $$x… Click here to read more

Let $$A$$ be a subset of a topological space $$X$$, a point $$x \in X$$ is said to be boundary… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then the sub collection $${\rm B} $$ of $$\tau $$ is… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space. A sub-collection $$S$$ of subsets of $$X$$ is said to be… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space and $$x \in X$$, then the sub collection $${{\rm B}_x}$$ is… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then $$X$$ is said to be the first countable space if… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then $$X$$ is said to be the second countable space, if… Click here to read more

Open Cover Let $$\left( {X,\tau } \right)$$ be a topological space. A collection $$\left\{ {{U_\alpha }:\alpha \in I} \right\}$$ of… Click here to read more

A topological space $$\left( {X,\tau } \right)$$ is said to be a separable space if it has a countable dense… Click here to read more

Let $$f$$ be a function defined from topological space $$X$$ to topological space $$Y$$, then $$f$$ is said to be… Click here to read more

Open Mapping A mapping $$f$$ from one topological space $$X$$ into another topological space $$Y$$ is said to be an… Click here to read more

A function $$f:X \to Y$$ is said to be a homeomorphism (topological mapping) if and only if the following conditions… Click here to read more

A property $$P$$ is said to be a topological property if whenever a space $$X$$ has the property $$P$$, all… Click here to read more

Products of Sets If $${X_1}$$ and $${X_2}$$ are two non-empty sets, then the Cartesian product $${X_1} \times {X_2}$$ is defined… Click here to read more

A topological space $$X$$ is said to be a $${T_o}$$ space if for any pair of distinct points of $$X$$,… Click here to read more

A topological space $$X$$ is said to be a $${T_1}$$ space if for any pair of distinct points of $$X$$,… Click here to read more

A Hausdorff space is a topological space in which each pair of distinct points can be separated by a disjoint… Click here to read more

Let $$\left( {X,\tau } \right)$$ be a topological space, then for every non-empty closed set $$F$$ and a point $$x$$… Click here to read more

A topological space $$X$$ is said to be a completely regular space if every closed set $$A$$ in $$X$$ and… Click here to read more

Let $$X$$ be a topological space and $$A$$ and $$B$$ are disjoint closed subsets of $$X$$, then $$X$$ is said… Click here to read more

A topological space $$X$$ is said to be a disconnected space if $$X$$ can be separated as the union of… Click here to read more

A topological space which cannot be written as the union of two non-empty disjoint open sets is said to be… Click here to read more

A connected subspace of a topological space $$X$$ is said to be the component of $$X$$ if it is not… Click here to read more

A topological space $$X$$ is said to be a totally disconnected space if any distinct pair of $$X$$ can be… Click here to read more

In some applications of connectedness, we shall define two fixed point theorems in connection with the application of connectedness. Fixed… Click here to read more

Cover and Sub-Cover Let $$X$$ be a topological space. A collection $$\left\{ {{A_\alpha }:\alpha \in I} \right\}$$ of subsets of… Click here to read more

A collection $$A$$ of subsets of a non-empty set $$X$$ is said to have the finite intersection property if every… Click here to read more

A space $$X$$ is said to be locally compact (briefly $$L – $$ Compact) at $$x \in X$$ if and… Click here to read more