Topological space

In mathematics, a topological space is an ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,O)} where $X$ is a set and $O$ is a certain collection of subsets of $X$ called the open sets or the topology of $X$. The topology of $X$ introduces a structure on the set $X$ which is useful for defining some important abstract notions such as the "closeness" of two elements of $X$ and convergence of sequences of elements of $X$.

Formal definition

A topological space is an ordered pair $(X,O)$ where $X$ is a set and $O$ is a collection of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} (i.e., any element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in O } is a subset of X) with the following three properties:

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} (the empty set) are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O}
2. The union of any number (countable or uncountable) of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O}
3. The intersection of any finite number of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O}

Elements of the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} are called open sets (of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ).

A topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,O)} is often simply written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} once the particular topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is understood.

Examples

1. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{R}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.}

Then a topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} can be defined on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{R}} to consist of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} and all sets of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is any arbitrary index set, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{\gamma}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{\gamma}} are real numbers satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_\gamma < b_\gamma} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \in \Gamma } . This is the familiar topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and in the next example another topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , albeit a relatively obscure one, will be constructed.

2. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{R}} as before. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} be a collection of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} defined by the requirement that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in O } if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\emptyset} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} contains all except at most a finite number of real numbers. Then it is straightforward to verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} defined in this way has the three properties required to be a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} . This topology is known as the cofinite topology or Zariski topology.

3. Every metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} on $X$ gives rise to a topology on $X$. The open ball with centre $x\in X$ and radius $r>0$ is defined to be the set

$B_{r}(x)=\{y\in X\mid d(x,y)<r\}.$

A set $A\subset X$ is open if and only if for every $x\in A$, there is an open ball with centre $x$ contained in $A$. The resulting topology is called the topology induced by the metric $d$. The standard topology on $\mathbb {R}$, discussed in Example 1, is induced by the metric $d(x,y)=|x-y|$.

Neighbourhoods

Given a topological space $(X,O)$, we say that a subset N of X is a neighbourhood of a point $x\in X$ if N contains an open set $U\in O$ containing the point x.

In this article, we defined topological spaces in terms of their open sets. It is also possible to define topological spaces in terms of neighbourhoods. In this alternative approach, a topological space is defined by the set $X$ and for each $x\in X$, a collection $N_{x}$ of subsets of $X$ (the neighbourhoods of $x$), such that:

1. $N_{x}$ is not empty for any $x\in X$
2. If $U$ is in $N_{x}$ then $x\in U$
3. The intersection of two elements of $N_{x}$ is again in $N_{x}$
4. If $U$ is in $N_{x}$ and $V\subset X$ contains $U$, then $V$ is again in $N_{x}$
5. If $U$ is in $N_{x}$ then there exists a $V\in N_{x}$ such that $V$ is a subset of $U$ and $V\in N_{y}$ for all $y\in V$

The open sets in this topology are precisely those sets $A$ that are in $N_{x}$ for all $x\in A$.

This alternative approach is particularly useful when one considers topologies on topological abelian groups and topological rings by subgroups or ideals, respectively, because knowing the neighbourhoods of any point is equivalent to knowing the neighbourhoods of 0.

Some topological notions

This section introduces some important topological notions. Throughout, X will denote a topological space with the topology O.

Partial list of topological notions

Limit point

A point $x\in X$ is a limit point of a subset A of X if any open set in O containing x also contains a point $y\in A$ with $y\neq x$. An equivalent definition is that $x\in X$ is a limit point of A if every neighbourhood of x contains a point $y\in A$ different from x.

Open cover

A collection ${\mathcal {U}}$ of open sets of X is said to be an open cover for X if each point $x\in X$ belongs to at least one of the open sets in ${\mathcal {U}}$

Path

A path $\gamma$ is a continuous function $\gamma :[0,1]\rightarrow X$. The point $\gamma (0)$ is said to be the starting point of $\gamma$ and $\gamma (1)$ is said to be the end point. A path joins its starting point to its end point

Hausdorff/separability property

X has the Hausdorff (or separability) property if for any pair $x,y\in X$ there exist disjoint sets U and V with $x\in U$ and $y\in V$

Connectedness

X is connected if given any two disjoint open sets U and V such that $X=U\cup V$, then either X=U or X=V

Path-connectedness

X is path-connected if for any pair $x,y\in X$ there exists a path joining x to y

Compactness

X is said to be compact if any open cover of X has a finite sub-cover. That is, any open cover has a finite number of elements which again constitute an open cover for X

A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.

Induced topologies

A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as induced topologies. Descriptions of some induced topologies are given below. Throughout, $(X,O_{X})$ will denote a topological space.

Some induced topologies

Relative topology

If A is a subset of X then open sets may be defined on A as sets of the form $O\cap A$ where O is any open set in $O_{X}$. The collection of all such open sets defines a topology on A called the relative topology of A as a subset of X

Quotient topology

If Y is another set and q is a surjective function from X to Y then open sets may be defined on Y as subsets U of Y such that $q^{-1}(U)=\{x\in X\mid q(x)\in U\}\in O_{X}$. The collection of all such open sets defines a topology on Y called the quotient topology induced by q

See also

• Topology
• Analysis
• Metric space

References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
2. D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [1]