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Point-set topology is a fundamental branch of topology, sometimes referred to as general topology, which deals with the concepts of topological spaces and the mathematical structures defined on such spaces.

## Topology and open sets

Given a set $X$ , a family of subsets $\tau$ of $X$ is said to be a topology of $X$ if the following three conditions hold:

1. $X,\varnothing\in\tau$ (The empty set and $X$ are both elements of $\tau$)
2. $\{A_i\}_{i\in I}\in\tau\rArr\bigcup_{i\in I}A_i\in\tau$ (Any union of elements of $\tau$ is an element $\tau$)
3. $A,B\in\tau\rArr A\cap B\in\tau$ (Any finite intersection of elements of $\tau$ is an element of $\tau$)

The members of a topology are called open sets of the topology.

## Topological space

A topological space is a set $X$ , known as the underlying set, together with a topology T of $X$ .

## Basis for a topology

A basis for a topology on $X$ is a collection of subsets of $X$ , known as basis elements, such that the following two properties hold:

1. For every $x\in X$ there is at least one basis element $B$ that contains $x$ .
2. If $x$ is an element of the intersection of two basis elements $A,B$ , then there exists a basis element $C$ such that $C\subset A\cap B$ .

Given a basis for a topology, one can define the topology generated by the basis as the collection of all sets $A$ such that for each $x\in A$ there is a basis element $B$ such that $x\in B$ and $B\subset A$.

## Closed sets

A set $C$ is defined to be closed if its complement in $X$ is an open set in the given topology.

## Neighborhoods

A set $N$ is said to be a neighborhood of a point $a$ if it is an open set which contains the point $a$ . In some cases the term neighborhood is used to describe a set which contains an open set containing $a$ .

## Interior and closure

The interior of a subset $A$ of $X$ is defined to be the union of all open sets contained in $A$ .

The closure of a subset $A$ of $X$ is defined as the intersection of all closed sets containing $A$ .

## Limit points

A point $x$ of $X$ is said to be a limit point of a subset A of $X$ if every neighborhood of $x$ intersects A in at least one point other than $x$ .

## Continuous functions

A function $f:X\to Y$ is said to be continuous if for each subset $A$ of $Y$ , the set $f^{-1}(A)$ is an open set of $X$ .

## Homeomorphisms

A bijective function $f:X\to Y$ is said to be a homeomorphism if both $f$ and its inverse, $f^{-1}:Y\to X$ , are continuous.

If there exists a homeomorphism between two topological spaces X and Y, then the spaces are said to be homeomorphic.

Any property that is invariant under homeomorphisms is known as a topological property.

A homeomorphism is also dubbed a topological equivalence among mathematicians.

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