## 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

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 A collection of subsets of which can be can be expressed as a 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 be 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 be a topological space with topology , and be 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 be 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 the neighborhood of if there exists an... 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 a base or bases or open... Click here to read more

Let be a topological space. A sub-collection of subsets 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 the first countable space if for every has a countable... Click here to read more

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

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

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

Let be a function defined 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 a homeomorphism (topological mapping) if and only if the following conditions are satisfied: (1)... 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 the Cartesian product is defined as . Projection Maps... Click here to read more

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

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

A topological space is said to be a 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 a disconnected space if can be separated as the union of two non-empty... 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 a totally disconnected space if any distinct pair 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 the application of connectedness. Fixed... 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