Point-Set Topology
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 54 (General topology)

Prerequisites: set theory, real analysis

# Getting Oriented

 Rough Guides to Topology
 General Topology Algebraic Topology Manifold Theory Knot Theory

Point-set topology is the study of the intrinsic properties of surfaces that are independent of distance. The classic example is the donut and the coffee cup, which, from our point of view, will be the same object.

We begin by looking at metric spaces, for which distance is defined, allowing the definition of an open sets as a bunch of points which are “close together”. Then we will take away the metric (distance), and just look at a set of points. How can we tell how those points are connected without distance? Well, we just skip the metric and begin by defining which sets of points are open (close together). This collection of open sets is what we mean by a topology.

Perhaps the most important concept here is the notion of homeomorphic topological spaces, which are just those that have the same topology… the underlying points are connected in the same manner (back to the donut and the coffee cup). We look at several topological properties that are preserved under this kind of equivalence. Of particular importance are the ideas of compact and connected spaces, which allow for generalizations of the Intermediate Value Theorem and the Extreme Value Theorem for real-valued functions.

# Metric Spaces

In metric spaces (top drawing), a distance is defined between any two points. This is what gives the space the notion of "closeness". Balls are defined as the set of points within a certain distance of a particular point.

In topological spaces (bottom drawing), there is no notion of distance, and the notion of proximity is captured by open sets: two points are "close" if they are "usually in the same open set".

A metric space is a set of points whose only structure is a notion of distance.

Definition

Metric Space : a set $X$ of points together with a distance function with $d: X\times X \to \mathbb{R}$ such that:

1. $d(x,y)\geq 0$ and $d(x,y)=0 \iff x=y$ (positivity);
2. $d(x,y)=d(y,x)$ (symmetry);
3. $d(x,z)\leq d(x,y)+d(y,z)$ (triangle inequality).

A typical example is the real numbers $\mathbb{R}$ with distance $d(x,y)=|x-y|$, or Euclidean space $\mathbb{R}^n$ with $d(\mathbf{x},\mathbf{y})=\sqrt{(x_1-y_1)^2+\cdots+(x_n-y_n)^2}$. Another example is a graph, in which the distance between nodes in the graph is the size of the minimum path between them.

The basic open set in $X$ is the open ball $B_\epsilon(x)=\{y\in X:d(x,y)<\epsilon\}.$ A metric space is bounded if it can be covered by a single $\epsilon$-ball. It is totally bounded if it can be covered by a finite number of $\epsilon$-balls for any $\epsilon>0$. The diameter of a bounded set is the least upper bound of the distances between two points in the set. The distance between two sets is the greatest lower bound of the distances between points in the sets.

A sequence $x_1, x_2, x_3, \ldots$ in $X$ is Cauchy if given any $\epsilon>0$ there is an $N>0$ such that $d(x_n,x_m)<\epsilon$ for all $n,m>N$. Complete spaces are those in which every Cauchy sequence converges.

# Topological Spaces

Closely related to metric spaces are topological spaces, which are equipped with a notion of closeness without the need for a distance function. In the topological approach, the notion of an open set is used to "measure" closeness: two points are "close" if they share in common "most" of their open sets.

Definition

Topological Space : a set of points $X$ together with a collection of subsets $\mathcal{T}$, which are by definition the open subsets of $X$, such that

1. any union of open sets is open (may be infinite);
2. any finite intersection of open sets is open;
3. the set $X$ and the empty set $\emptyset$ are both open.

The collection $\mathcal{T}$ of subsets is known as a topology for $X$. An open set $U\in\mathcal{T}$ containing a point $x$ is called a neighborhood of $x$.

Topologies may be compared to each other. If all open sets in a topology $\mathcal{T}_1$ are also open in $\mathcal{T}_2$, then $\mathcal{T}_2$ is finer (or sometimes larger or stronger) than $\mathcal{T}_1$, while $\mathcal{T}_1$ is coarser (or smaller or weaker) than $\mathcal{T}_2$. The coarsest topology is the trivial topology $\mathcal{T}_{trivial}=\{\emptyset,X\}$, while the finest topology is the discrete topology $\mathcal{T}_{discrete}$, in which every subset of $X$ is defined to be open.

One of the most useful tricks for proving a set $U$ is open is the following:

Theorem

A set $U$ is open if and only if for every $x\in U$, there is a neighborhood $V$ such that $x\in V\subset U$.

## Closed Sets

The topology also defines the closed sets:

Definition

Closed Set : a set $A\in X$ whose complement $A^c$ is open, i.e., $A^c\in\mathcal{T}$.

Infinite unions of closed sets may not be closed, and infinite intersections of open sets may not be open.

One can show the following facts about closed sets using deMorgan's Laws:
• Any intersection of closed sets is closed (may be infinite);
• Any finite union of closed sets is closed.

The standard topology on Euclidean space $\mathbb{R}^n$ is formed from the unions of open balls. In $\mathbb{R}$, the basic open sets have the form $(a,b)$, while the basic closed sets have the form $[a,b]$, for $a<b$. Here we can see that arbitrary unions of closed sets may not be closed, just as arbitrary intersections of open sets may not be open. For example,

(1)
\begin{eqnarray} \bigcap_{i=1}^\infty \left(-\frac{1}{i}, \frac{1}{i}\right) &=& \{0\} \\ \bigcup_{i=1}^\infty \left[-1+\frac{1}{i}, 1-\frac{1}{i}\right] &=& (-1,1). \end{eqnarray}

## Basis of a Topology

When it's not practical to give a list of all open sets, one might use a basis or subbasis for the topology. A basis is a collection of open sets whose unions give the topology, while a subbasis is a collection of open sets whose unions and finite intersections give the topology. In a metric space, the set of open balls forms a basis for the metric topology. The set of open balls itself is not a topology since unions of open balls are open but may not themselves be open balls. The metric topology coincides with the standard topology on Euclidean space.

Many facts about topologies may be proven by verifying them for the basis rather than the entire collection of open sets. If one wishes to prove that a topology $\mathcal{T}_1$ is finer than a second topology $\mathcal{T}_2$, it suffices to show that every basis element $B_2$ of $\mathcal{T}_2$ is open in $\mathcal{T}_1$. By a result mentioned earlier, this may be proven by finding basis elements of $\mathcal{T}_1$ about every point in $B_2$.

A neighborhood basis around a point $x\in X$ is a (possibly infinite or uncountable) collection of neighborhoods $\{B_\alpha\}$ of that point such that any neighborhood of $x$ is contained in some $B_\alpha$. The topological space $X$ is first countable if every point has a countable neighborhood basis, and second countable if the topology has a countable basis.

# Functions between Topological Spaces

For a continuous function, preimages of open sets are open, and preimages of closed sets are closed.

As shown by this example, f(x)=x2, the forward image of an open set might not be open.

In topology, continuity is expressed in terms of open sets:

Definition

Continuous Function (or Map) : a function $f:X\to Y$ between topological spaces such that $f^{-1}(U)$ is open whenever $U\subset Y$ is open. A function is continuous at a point $x\in X$ if it is continuous on a neighborhood of $x$.

Note the "directionality" of this definition: one needs the inverse image of sets to be open, rather than the forward image. There is another name for the second case: a function $f:X\to Y$ is an open (closed) function if $f(A)\subset Y$ is open (closed) whenever $A\subset X$ is open (closed).

Continuity depends very much on the underlying topology. For the standard topology on $\mathbb{R}^n$, continuity is equivalent to the standard epsilon-delta definition. Continuity can be a trivial condition: if $X$ has the discrete topology and/or $Y$ has the trivial topology, then any function from $X$ to $Y$ is continuous.

Some properties of sets may be expressed in terms of continuous functions. For example, a space $X$ is connected if and only if every discrete-valued map on $X$ is constant. (Discrete-valued means the function maps into a space with the discrete topology.) Similarly, we will later see that certain topologies are defined as precisely those which make specific maps continuous.

Continuous maps are those which, in some sense, maintain some form of topological structure on a space. If the topological structure is the same for both $X$ and $Y$, we say the spaces are homeomorphic. Proving this property requires finding a homeomorphism:

Definition

Homeomorphism : a bijective map $f:X\to Y$ with both $f$ and $f^{-1}$ continuous.

For example, the open interval $(-1,1)$ is homeomorphic to all of $\mathbb{R}$, via the homeomorphism $f(x)=\frac{x}{|x|+1}$. Similarly, the open disk $\left\{(x,y):x^2+y^2<r\right\}$ in $\mathbb{R}^2$ is homeomorphic to all of $\mathbb{R}^2$.

Homeomorphisms also exist between topological spaces represented in very different ways. The torus (or "donut") in $\mathbb{R}^3$ is homeomorphic to a unit square with opposite sides "glued together". These constructions are discussed in the next section.

# Constructing Topological Spaces

## The Subspace Topology

Open sets in the subspace topology are formed by intersections of the set with open sets in the base topology.

All subsets $A\subset X$ inherit a topology from $X$, defined by intersecting the open sets in $X$ with $A$. Thus, one may speak about the standard topology on the unit square $[0,1]\times[0,1]$ in $\mathbb{R}^2$, or the topology on an interval $(0,1)$, or the topology on a surface in $\mathbb{R}^3$. These are assumed to "inherit" their topologies from the ambient space.

One may also define a topology (the obvious one) on a disjoint union $X\sqcup Y$: sets $U\sqcup V\subset X\sqcup Y$ are open if and only if $U\subset X$ and $V\subset Y$ are both open.

## The Product Topology

Open sets in the product topology are unions of products of open sets in the base topologies.

The Cartesian product of spaces $X$ and $Y$ is the set of points $X\times Y = \left\{(x,y) : x\in X, y\in Y\right\}$. The topologies of $X$ and $Y$ can be used to construct a topology on the product space $X\times Y$. The idea is

Definition

Finite Product Topology : the topology on $\prod_{i=1}^n X_i$ whose basis consists of sets of the form $\prod_{i=1}^n U_i$, where each $U_i \subset X_i$ is open.

This definition extends to arbitrary products $\prod_\alpha X_\alpha$, but if the product is infinite this construction may not give the "right" topology in the sense that it has "too many open sets". The distinction is subtle, with the infinite product called the box topology in this case. The "right" construction is the product topology (or Tychonoff topology), which is the coarsest topology such that the projection maps $\prod X_\alpha \to X_\alpha$ are all continuous. It's basis consists of sets of the form $\prod_\alpha U_\alpha$, where $U_\alpha = X_\alpha$ for all but finitely many $\alpha$.

If $X_\alpha=X$ for all $\alpha \in A$, the product space is $X^A$, the space of functions $A\to X$.

## The Quotient Topology

In topology, quotients of spaces are obtained by "gluing points together". It might be better known as the "gluing topology".

Definition

Quotient space : the space $X/\sim$, where $\sim$ is an equivalence relation on $X$, with topology defined so that the inclusion $X/\sim \hookrightarrow X$ is continuous. In other words, a set is open in $X/\sim$ if and only if its preimage in $X$ is open.

The quotient of the interval obtained by gluing endpoints together is homeomorphic to the circle S1.

The topology on the quotient space is the quotient topology. Sets in the quotient space are said to descend from sets in $X$.

An alternate version of the same construction is to define the quotient topology on $Y$ using a surjection $f:X\to Y$. Again, $U\subset Y$ is open if and only if $f^{-1}(U)$ is open, so that the topology is defined by requiring the surjection to be continuous. The equivalence relation corresponding to the surjection is $x_1\sim x_2$ whenever $f(x_1)=f(x_2)$.

An example is the interval $[0,1]$ with endpoints glued together, so that $0\sim 1$; this is topologically equivalent to the unit circle. Another example is the unit square $[0,1]\times[0,1]$ with opposite sides identified; as indicated earlier, this is topologically equivalent to the torus.

## The Topological Zoo

Many well-known topological spaces are constructed as quotient spaces of subsets of Euclidean space. Two non-homeomorphic spaces may be constructed by gluing opposite sides of a square.

An annulus (or a cylinder) is the quotient of a square obtained by identifying opposite sides.

There are more possibilities if one glues all sides of a square in some way. Two of the resulting spaces are the torus and the Klein bottle.

A torus (or donut) is obtained by gluing opposite sides of a square together.

# Connectedness, Limit Points, and Separations

## Connectedness

In general, subsets may be open, closed, both open and closed (called clopen), or neither open nor closed. The extreme sets $\emptyset$ and $X$ are always clopen. The notion of open and closed sets also permits a description of when a set consists of "one big piece" rather than several smaller pieces:

Definition

Connected space : a space $X$ which is not the disjoint union of two nonempty open subsets. Equivalently, the only clopen sets are $\emptyset$ and $X$.

A separation of $X$ consists of two disjoint nonempty open sets whose union is $X$. The components of $X$ are the largest connected subsets of $X$. A more intuitive definition of "connected" is known as path-connected, and is determined by whether any two points are connected by a path. This notion is not quite the same as the above definition, and will be discussed later.

If $f:X\to Y$ is continuous, and $A\subset X$ is connected, then $f(A)$ is also connected. The converse may not hold. This implies the following important property of real-valued functions:

Intermediate Value Theorem

A real-valued map $f:X\to\mathbb{R}$ on a connected space X has the intermediate-value property: if $p,q\in f(X)$ then f maps some point to every value between p and q.

A fixed-point of a function $f:X\to X$ is a point $x\in X$ with $f(x)=x$. The simplest fixed-point theorem depends directly on the Intermediate Value Theorem:

Brouwer Fixed-Point Theorem (Dimension 1)

Every continuous function $f:[0,1]\to[0,1]$ has a fixed point.

The general Brouwer Fixed-Point Theorem states that any function $f:[0,1]\times\cdots\times[0,1]\to[0,1]\times\cdots\times[0,1]$ has a fixed point.

## Sequences and Limits

In the most common cases, open sets contain sequences whose limits may be outside the set (top). In contrast, closed sets contain all their limit points (bottom).

A limit point of a subset $A\subset X$ is a point $x\in X$ such that every neighborhood of x intersects A.
One can prove that closed sets contain all of their limit points.

Closely related, a sequence in a topological space is a set of points $x_1, x_2, \ldots$. A limit of the sequence is a point $x\in X$ such that every neighborhood of $x$ contains a tail of the sequence, that is $\{x_N,x_{N+1},\ldots\}$ is in the neighborhood for some $N$. This is equivalent to the standard $\epsilon-N$ definition in metric spaces.

Subsets may be "grown" to form a closed set, or "shrunk" to form an empty set. The closure $\mathrm{Cl}(A)$ or $\overline{A}$ of a set $A \subset X$ is the set of limit points of $A$, or equivalently the intersection of all closed sets containing $A$. The interior $\mathrm{Int}(A)$ or $\overset\circ{A}$ is the union of all open sets contained in $A$. The boundary $\partial A$ is the difference between these: $\partial A = \mathrm{Cl}(A) \setminus \mathrm{Int}(A)$.

The set A is dense in X if $\mathrm{Cl}(A)=X$, and nowhere dense if $\mathrm{Int}(\mathrm{Cl}(A))=\emptyset$.

## The Separation Axioms

In topology, "separation" indicates the ability of open sets to distinguish points. The separation axioms (sometimes labeled as $T_0, T_1, \ldots$) classify a topological space $X$ based on how easy it is to "separate" points into different subsets of $X$. There are various ways to formalize this kind of condition:
• $T_0$: given distinct points $x,y\in X$, there is an open set $U\ni x$ with $y\not\in U$ OR an open set $V\ni y$ with $x\not\in V$;
• $T_1$: given distinct points $x,y\in X$, there are open sets $U\ni x$ and $V\ni y$ with $y\not\in U$ and $x\not\in V$ (but $U$ and $V$ might overlap);
• $T_2$: given distinct points $x,y\in X$, there are disjoint open sets $U\ni x$ and $V\ni y$;
• $T_3$: given a point $x\in X$ and a closed set $F\subset X$ that does not contain $x$, there are disjoint open sets $U\ni x$ and $V\supset F$;
• $T_4$: given disjoint sets $F,G\in X$, there are disjoint open sets $U\supset F$ and $V\supset G$.
• $T_5$: given sets $F,G\in X$ for which $Cl(F)\cap G=F\cap Cl(G)=\emptyset$ (they are then said to be separated), there are disjoint open sets $U\supset F$ and $V\supset G$.

These conditions are listed in roughly order of increasing strength, although there are some exceptions. It is obvious that $T_2 \Rightarrow T_1 \Rightarrow T_0$. But the other implications are more complicated; for example, regular does not necessarily imply Hausdorff, since single points may not be closed.

These axioms may be strengthened, however, to ensure the "obvious" implications do hold. The strengthened (and named) versions are: Kolmogorov space (equivalent to $T_0$); Frechet space (equivalent to $T_1$); Hausdorff space (equivalent to $T_2$); regular space (both $T_3$ and $T_0$); normal space (both $T_4$ and $T_1$); completely normal space (both $T_5$ and $T_1$). Then

(2)
\begin{align} \text{completely normal} \Rightarrow \text{normal} \Rightarrow \text{regular} \Rightarrow \text{Hausdorff} \Rightarrow T_1 \Rightarrow T_0. \end{align}

As a word of warning, these lists are not exhaustive of all types of separation axioms. Also, various authors use reverse the meaning of regular and $T_3$, or use them interchangeably.

In general, $T_2$/Hausdorff is probably the most commonly used condition, since all metric spaces are Hausdorff. In geometry, manifolds are by definition assumed to be both Hausdorff and second countable (having a countable basis). This is related to two extremely desirable properties that hold in Hausdorff spaces: (i) single-point subsets are closed and (ii) convergent sequences have unique limits.

## Making Metrics

A topological space is said to be metrizable if there exists a metric on the space that induces the same topology. One of the key results in topology describes when this is possible:

Urysohn Metrization Theorem

A second countable regular space is metrizable.

The proof depends on two key results:
• Urysohn's Lemma: in a normal space, for any two closed/disjoint sets $C, D$ there is a function $f:X\to[0,1]$ with $f(C)=0$ and $f(D)=1$ (such a function is called an Urysohn function);
• Tietze Extension Theorem: a continuous map $f:C\to\mathbb{R}$ on a closed subset $C$ of a normal space can be extended to a map $\hat f:X\to\mathbb{R}$ on the entire space.

# Compactness

An open cover of a set consists of a finite or infinite collection of open sets whose union contains that set.

Because topological spaces have no notion of "distance", they do not really have a notion of "size" either. In some sense, however, the notion of compactness encapsulates the "small enough" topological spaces. Basically, the compact spaces allow a transformation from the infinite to the finite.

Given a subset $A\subset X$, an open cover of $A$ is a collection of open sets whose union contains $A$. A subcover is a subcollection of the cover that still contains $A$.

Definition

Compact set : a subset $A\subset X$ for which every open cover has a finite subcover.

In practice, the original cover is usually infinite, so that the defining property of compactness allows one to pass to a finite subcollection. There are a few different notions of compactness (to be discussed later). Simply "compact" generally refers to this particular definition, but the term covering compact is sometimes used for precision.

Examples of noncompact spaces. Both the open interval (a,b) and the infinite line $(-\infty,\infty)$ have open covers with no finite subcovers.

For example, $[a,b]$ is compact, while $(-\infty,b]$, $(a,b]$, and $(a,b)$ are not. In each of the latter cases, one can find an infinite open cover with no finite subcover. Another example is the unit square $[0,1]\times[0,1]\subset\mathbb{R}^2$. This is covered by open balls of radius $\epsilon$ for a fixed $\epsilon$. One can always find a finite number of such balls that also covers the set; hence it is compact.

An equivalent characterization of compact in terms of closed sets is the following: if $\mathcal{C}$ is a collection of closed sets such that all finite subcollections have nonempty intersection, then the entire collection has nonempty intersection.

If $f:X\to Y$ is continuous, and $A\subset X$ is compact, then $f(A)$ is also compact. When the converse holds, so that $f^{-1}(C)$ is compact for all compact $C\subset Y$, the map $f$ is called proper. Using the preservation of compact sets under continuity, one may prove the following fact essential for optimization problems in basic calculus:

Extreme Value Theorem

A real-valued map $f:X\to\mathbb{R}$ from a compact space $X$ assumes a maximum value.

The 1-point compactification of the open interval and the infinite line is homeomorphic to S1. The 1-point compactification of an open disk in R2 is homeomorphic to the sphere S2.

The following properties of compact spaces are very useful:
• closed subsets of a compact space are compact;
• the product of two compact sets is compact in the product space;
• a closed subset of a Hausdorff space is compact;
• a compact Hausdorff space is normal.

A space is locally compact if every point has a compact neighborhood. A locally compact Hausdorff space is completely regular. On such a space, one can define the 1-point compactification $X^+ = X\cup\{\infty\}$ by adjoining a point at infinity and asserting that $C^c\cup\{\infty\}$ is open for all compact $C$. The Hausdorff condition ensures that the result is a compact space, while the locally-compact condition ensures that the result remains Hausdorff.

## Compactness in Metric Spaces

In Euclidean space, compactness is easily characterized:

Heine-Borel Theorem

A subset of Euclidean space is compact if and only if it is closed and bounded.

This theorem also generalizes to general metric spaces: a subset of a metric space is compact if and only if it is complete (every Cauchy sequence converges) and totally bounded (can be covered by a finite number of $\epsilon$-balls for any $\epsilon>0$).

A nice feature of compact metric spaces is the Lebesgue Lemma, which states that for any open cover of a compact metric space, there exists some $\delta>0$ for which any set of diameter less than $\delta$ sits inside a single set of the cover. The value $\delta$ is called the Lebesgue number of the cover.

A space is sequentially compact if every sequence has a convergent subsequence. A space is limit point compact (or sometimes weakly-countable compact) if every infinite subset has a limit point. Covering compact and sequentially compact sets are limit point compact, but otherwise the conditions are generally exclusive. However, in metric spaces, the three conditions are the same: covering compact is equivalent to both sequentially compact and limit point compact.

# Going Further

## Path-Connectedness

A path in a topological space X is a (continuous) map $f:[0,1]\to X$. A space is path-connected if there exists a path between any two points in the space. The path components in a space are the largest path-connected pieces of the space.

Path-connected is a stronger property than connectedness, since paths are connected subsets of X. The topologist's whirlpool shown here is one example showing that connected does not imply path-connected.

## Homotopy

Fields that build upon point-set topology, such as algebraic topology, make extensive use of the idea that two maps from one topological space into another can sometimes be "continuously deformed" into each other, much like a piece of a string can be moved across a table. The notion that captures this idea is homotopy:

Definition

A homotopy between functions $f:X\to Y$ and $g:X\to Y$ is a continuous function $F:X\times I\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$. Then f and g are homotopic and we write $f\simeq g$.

One often thinks of the homotopy as parametrizing a family of functions $F_t:X\to Y$ for $0\le t\le 1$. Then, as t changes from 0 to 1, the homotopy transforms from $f(x)$ to $g(x)$. The homotopy class $[f]$ of f is the set of maps homotopic to f. Two homotopies $F$ and $G$ may be concatenated if $F_1=G_0$. The resulting homotopy, requiring a reparametrization, is denoted $F*G$.

Let $1_X$ denote the identity map on X. Two spaces X and Y are homotopy equivalent if there exist maps $f:X\to Y$ and $g:X\to Y$ with $g\circ f \simeq 1_X$ and $f\circ g \simeq 1_Y$ (that is, the composite maps are both homotopic to the identity), and we write $X\simeq Y$.

A space is contractible if it is homotopy equivalent to a 1-point space. A subset $A\subset X$ is a deformation retract of X if there is a homotopy F with $F_0=1_X$ the identity on X and $F_1(X)$ a retract from X to A. The subset A is a strong deformation retract if $F_t|_A=1_A$ for all $t$. In either case, $A\simeq X$.

A relative homotopy stays fixed on some set, i.e., $F_t|_A=1_A$, and is denoted by $F_0\simeq_A F_1$. Homotopies fixed on the endpoints $X_{01}=X\times\partial I$ are very important, since they satisfy some nice properties:

1. $C*F\simeq F*C\simeq_{X_{01}} F$ for any constant map C;
2. every F has an inverse $F^{-1}$ with $F*F^{-1}\simeq_{X_{01}}1_X$;
3. if $F_1,G_1\simeq F_2,G_2$ then $F_1*G_1\simeq_{X_{01}} F_2*G_2$.

So if $[e]$ is the homotopy class of the identity, then $[e]*[f]=[f]*[e]=[f]$, $[f]*[f^{-1}]=[e]$, and $[f]*[g]=[fg]$ is well-defined. But these are precisely the axioms of a group! Such homotopy groups are the starting point of algebraic topology and the fundamental group.

The theory here does not yet answer the question of how to distinguish and classify topological spaces, a fundamental goal given any mathematical structure. But one tool which can do just this is present: the homotopy group. Indeed, the simplest tool which is used to study differences between topological spaces is perhaps the fundamental group, which is the set of maps from the circle $S^1$ into a topological space, up to homotopy (a basepoint for each such map must also be specified). The group is invariant under homeomorphism, and so can be used to distinguish spaces. Knot theory also draws heavily on point-set topology, in particular looking at embeddings of loops in 3-space.