Topological space

In mathematics, a topological space is an ordered pair $(X,O)$ 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 $X$.

Definition

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

1. $X$ and $\emptyset$ (the empty set) are in $O$
2. The union of any family (infinite or otherwise) of elements of $O$ is again in $O$
3. The intersection of finitely many elements of $O$ is again in $O$

Elements of the set $O$ are called open sets of $X$. We often simply writte $X$ instead of $(X,O)$ once the topology $O$ is established.

Once we have a topology of open sets on $X$, we define the closed sets of $X$ to be the compliments (in $X$) of the open sets; the closed sets of $X$ have the properties that

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

Alternatively, notice that we could have defined a topology of closed sets (having the properties above as axioms) and defined the open sets relative to that topology as compliments of closed sets. The set of opens relative to a topology of closed sets then obeys the axioms for a topology of opens, and therefore closed and these two kinds of topologies occur simultaneously. Similarly, the axioms for a topology of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology of opens, and conversely. Thus, the notion of topology is a relative one, and these equivalences allow one to study the same topological structure from a number of different viewpoints simultaneously.

The category of topological spaces

Given that topological 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 $O$ 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 $O$ be a collection of subsets of $\mathbb {R}$ defined by the requirement that $A\in O$ 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 $O$ 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 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 r > 0} is defined to be 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 B_r(x) = \{ y \in X \mid d(x,y) < r \}. }

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

Neighborhoods

Given 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)} of opens, we say that a subset 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 N} 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} is a neighborhood of a point 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 \in X} 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 N} contains an open 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 U \in O} containing the point 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}

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 N_x} denotes the system of neighborhoods 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} relative to the 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} , then the following properties hold:

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 N_x} is not empty for any 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 \in X}
2. 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 U} is 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 N_x} then 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 \in U}
3. The intersection of two 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 N_x} 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 N_x}
4. 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 U} is 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 N_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 V \subset X} contains 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 U} , then 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 V} 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 N_x}
5. 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 U} is 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 N_x} then there exists a 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 V \in N_x} such 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 V} is a subset 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 U} 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 V \in N_y} 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 y \in V}

Conversely, if we define a topology of neighborhoods 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} 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: 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 U} is open if it is 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 N_x} 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 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 spaces 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 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 I} -adic topology on a ring 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} is Hausdorff 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 \bigcap I^n=0} , thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.

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

Open cover

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

Path

A path 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 a continuous function 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:[0,1]\rightarrow X} . The point 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(0)} is said to be the starting point 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 \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 \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 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,y \in X} there exist disjoint sets U and V with 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 \in U} 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 y \in V}

Connectedness

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

Path-connectedness

X is path-connected if for any 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,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, 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_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 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 \cap A} where O is any open set 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_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 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 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]