Topological space

In mathematics, a topological space is an ordered pair $(X,{\mathcal {T}})$ where $X$ is a set and ${\mathcal {T}}$ is a certain collection of subsets of $X$ called the open sets or the topology of $X$. The topology of $X$ introduces an abstract structure of space in the set $X$, which allows to define general notions such as of a point being surrounded by a set (by a neighborhood) or belonging to its boundary, of convergence of sequences of elements of $X$, of connectedness, of a space or set being contractible, etc.

Definition

A topological space is an ordered pair $(X,{\mathcal {T}})$ where $X$ is a set and ${\mathcal {T}}$ is a collection of subsets of $X$ (i.e., any element $A\in {\mathcal {T}}$ is a subset of X) with the following three properties:

1. $X$ and $\varnothing$ (the empty set) are in ${\mathcal {T}}$
2. The union of any family (infinite or otherwise) of elements of ${\mathcal {T}}$ is again in ${\mathcal {T}}$
3. The intersection of two elements of ${\mathcal {T}}$ is again in ${\mathcal {T}}$

Elements of the set ${\mathcal {T}}$ are called open sets of $X$. We often simply write $X$ instead of $(X,{\mathcal {T}})$ once the topology ${\mathcal {T}}$ is established.

Once we have a topology in $X$, we define the closed sets of $X$ to be the complements (in $X$) of the open sets; the closed sets of $X$ have the following characteristic properties:

1. $X$ and $\varnothing$ (the empty set) are closed
2. The intersection of any family of closed sets is closed
3. The union of two closed sets is closed

Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as complements of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set $X$ we obtain a one to one correspondence between topologies in $X$ and systems of neighborhoods in $X$. These correspondences allow one to study the topological structure from different viewpoints.

The category of topological spaces

Given that topological spaces capture notions of geometry, a good notion of isomorphism in the category of topological spaces should require that equivalent spaces have equivalent topologies. The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.

A function $f:X\to Y$ is continuous if $f^{-1}(V)$ is open in $X$ for every open in $Y$. Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if $f^{-1}(Z)$ is closed in $X$ for each closed subset $Z$ of Y, then $f$ is continuous in the sense just defined, and conversely.

Isomorphisms in the category of topological spaces (often referred to as a homeomorphism) are bijective and continuous with continuous inverses.

The category of topological spaces has a number of nice properties; there is an initial object (the empty set), subobjects (the subspace topology) and quotient objects (the quotient topology), and products and coproducts exist as well. The necessary topologies to define on the latter two objects become clear immediately; if they're going to be universal in the category of topological spaces, then the topologies should be the coarsest making the canonical maps commute.

Examples

1. Let $X=\mathbb {R}$ where $\mathbb {R}$ denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:

${\mathopen {]}}a,b{\mathclose {[}}=\{y\in \mathbb {R} \mid a<y<b\}.$

Then a topology ${\mathcal {T}}$ can be defined on $X=\mathbb {R}$ to consist of $\emptyset$ and all sets of the form:

$\bigcup _{\gamma \in \Gamma }{\mathopen {]}}a_{\gamma },b_{\gamma }{\mathclose {[}},$

where $\Gamma$ is any arbitrary index set, and $a_{\gamma }$ and $b_{\gamma }$ are real numbers satisfying $a_{\gamma }<b_{\gamma }$ for all $\gamma \in \Gamma$. This is the familiar topology on $\mathbb {R}$ and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set $X$ and in the next example another topology on $\mathbb {R}$, albeit a relatively obscure one, will be constructed.

2. Let $X=\mathbb {R}$ as before. Let ${\mathcal {T}}$ be a collection of subsets of $\mathbb {R}$ defined by the requirement that $A\in {\mathcal {T}}$ if and only if $A=\emptyset$ or $A$ contains all except at most a finite number of real numbers. Then it is straightforward to verify that ${\mathcal {T}}$ defined in this way has the three properties required to be a topology on $\mathbb {R}$. This topology is known as the cofinite topology or Zariski topology.

3. Every metric $d$ 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 $d(x,y)=|x-y|$.

4. For a given set $X$, the family ${\mathcal {T}}=\{\emptyset ,X\}$ is a topology: the indiscrete or weakest topology.

5. For a given set $X$, the family ${\mathcal {T}}={\mathcal {P}}X$ of all subsets of $X$ is a topology: the discrete topology.

Neighborhoods

Given a topological space $(X,{\mathcal {T}})$ of opens, we say that a subset $N$ of $X$ is a neighborhood of a point $x\in X$ if $N$ contains an open set $U\in {\mathcal {T}}$ containing the point $x$

If $N_{x}$ denotes the system of neighborhoods of $x$ relative to the topology ${\mathcal {T}}$, then the following properties hold:

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 $U\in N_{y}$ for all $y\in V$

Conversely, if we define a topology of neighborhoods on $X$ via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from: $U$ is open if it is in $N_{x}$ for all $x\in U$. Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from. All this just means that a given topological space is the same, regardless of which axioms we choose to start from.

The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the $I$-adic topology on a ring $A$ is Hausdorff if and only if $\bigcap I^{n}=0$, thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.

Bases and sub-bases

A basis for the topology ${\mathcal {T}}$ on X is a collection ${\mathcal {B}}$ of open sets such that every open set is a union of elements of ${\mathcal {B}}$. For example, in a metric space the open balls form a basis for the metric topology. A sub-basis ${\mathcal {S}}$ is a collection of open sets such that the finite intersections of elements of ${\mathcal {S}}$ form a basis for ${\mathcal {T}}$.

Some topological notions

This section introduces some important topological notions. Throughout, $X$ will denote a topological space with the topology ${\mathcal {T}}$.

Partial list of topological notions

Closure

The closure in $X$ of a subset $E$ is the intersection of all closed sets of $X$ which contain $E$ as a subset.

Interior

The interior in $X$ of a subset $E$ is the union of all open sets of $X$ which are contained in $E$ as a subset.

Limit point

A point $x\in X$ is a limit point of a subset $A$ of $X$ if any open set in ${\mathcal {T}}$ 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, or T2) property if for any pair $x,y\in X$ there exist disjoint sets $U$ and $V$ with $x\in U$ and $y\in V$.

Noetherianity

$X$ is noetherian if it satisfies the descending chain condition for closed set: any descending chain of closed subsets $Y_{0}\supseteq Y_{1}\supseteq \ldots$ is eventually stationary; i.e., if there is an index $i\geq 0$ such that $Y_{i+r}=Y_{i}$ for all $r\geq 0$.

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 $y$. A path connected topological space is also connected, but the converse need not be true.

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.

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, $(X,{\mathcal {T}}_{X})$ 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 $U\cap A$ where $U$ is any open set in ${\mathcal {T}}_{X}$. 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 $q^{-1}(U)=\{x\in X\mid q(x)\in U\}\in {\mathcal {T}}_{X}$. The collection of all such open sets defines a topology on $Y$ called the quotient topology induced by $q$.

Product topology

If $(X_{\lambda },{\mathcal {T}}_{\lambda })_{\lambda \in \Lambda }$ is a family of topological spaces, then the product topology on the Cartesian product $\prod _{\lambda \in \Lambda }X_{\lambda }$ has as sub-basis the sets of the form $\prod _{\lambda \in \Lambda }U_{\lambda }$ where each $U_{\lambda }\in {\mathcal {T}}_{\lambda }$ and $U_{\lambda }=X_{\lambda }$ for all but finitely many $\lambda \in \Lambda$.

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]