## Definition of Topology

Let be a non empty set. A collection of subsets of is said to be a topology on if the... Click here to read more

From basic to higher mathematics

Let be a non empty set. A collection of subsets of is said to be a topology on if the... Click here to read more

If and are two topologies defined on the non empty set X such that , i.e. each member of is... 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

Intersection of any two topologies on a non empty set is always topology on that set. While the union of... Click here to read more

Usual Topology on : A collection of subsets of which can be can be expressed as union of open intervals,... Click here to read more

Let be a topological space, then a member of is said to be an open set in . Thus, in... Click here to read more

Let is a non empty set, and then the collection of subsets of whose compliments are finite along with (empty... Click here to read more

Let be a topological space, then a subset of X whose complement is a member of is said to be... Click here to read more

We shall describe a method of constructing new topologies from the given ones. If is a topological space and is... Click here to read more

Let is a topological space with topology , and is a subset of . A point is said to be... Click here to read more

Let be a subset of a topological space , then a point is said to be an isolated point of... Click here to read more

Let be a topological space and is a subset of , then the closure of is denoted by or is... Click here to read more

Let be a topological space. A subset of containing is said to be neighbourhood of if there exist an open... Click here to read more

Let be the topological space and , then a point is said to be an interior point of set ,... Click here to read more

Let be a topological space and be a subset of , then a point , is said to be an... Click here to read more

Let be a subset of a topological space , a point is said to be boundary point or frontier point... Click here to read more

Let be a topological space, then the sub collection of is said to be base or bases or open base... Click here to read more

Let be a topological space. A sub-collection of subset of is said to be an open subbase for or a... Click here to read more

Let be a topological space and , then the sub collection is said to be local bases at a point... Click here to read more

Let be a topological space, then is said to be first countable space if for every has a countable local... Click here to read more

Let be a topological space, then is said to be second countable space, if has a countable bases. In other... Click here to read more

Open Cover: Let be a topological space. A collection of open subsets of is said to be open cover for... Click here to read more

A topological space , is said to be separable space, if it has a countable dense subset in . i.e.,... Click here to read more

Let be a function define from topological space to topological space , then is said to be continuous at a... Click here to read more

Open Mapping: A mapping from one topological space into another topological space is said to be an open mapping if... Click here to read more

A function is said to be homeomorphism (topological mapping) if and only if the following conditions are satisfied: (1) is... Click here to read more

A property is said to be a topological property if whenever a space has the property , all spaces which... Click here to read more

Products of Sets: If and are two non-empty sets, then Cartesian product is defined as . Projection Maps: Let and... Click here to read more

A topological space is said to be a space if for any pair of distinct points of , there exist... Click here to read more

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

Let be a topological space, then for every non-empty closed set and a point which does not belongs to ,... Click here to read more

A topological space is said to be completely regular space, if every closed set in and a point , ,... Click here to read more

Let be a topological space and, and are disjoint closed subsets of , then is said to be normal space,... Click here to read more

A topological space is said to be disconnected space if can be separated as the union of two non-empty disjoint... 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 is said to be the component of if it is not properly contained... Click here to read more

A topological space is said to be totally disconnected space if any pair of distinct of can be separated by... Click here to read more

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

Cover and Sub-Cover: Let be a topological space. A collection of subsets of is said to be a cover of... Click here to read more

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

A space is said to be locally compact (briefly Compact) at if and only if has a compact neighbourhood in... Click here to read more