ACE 328/Chapter 1

Topological Spaces

The foundational structure of analysis: open sets, closed sets, and the topology axioms. Includes the discrete, trivial, and Euclidean topologies.

Real analysis, at its core, is the study of "closeness" — what it means for points to be near each other, for sequences to converge, and for functions to be continuous. In a first calculus course, closeness is measured by the absolute value $|x - y|$ on $\mathbb{R}$ ; in $\mathbb{R}^d$ , by the Euclidean norm $\|\boldsymbol{x} - \boldsymbol{y}\|$ . But these notions of distance are, in a sense, extra structure. The deeper structure underneath convergence and continuity is simply a choice of which subsets of the space count as "open."

That choice is called a topology. A topological space is the most general setting in which limits, continuity, and neighbourhood make sense. Before diving into metric spaces (where a distance function generates a topology automatically), we take the abstract viewpoint so that we understand which results require distance and which are purely topological.

Preliminaries: Power Sets and Set Notation

DefinitionPower Set

Let $X$ be a set. The power set of $X$ , denoted $2^X$ (or sometimes $\mathcal{P}(X)$ ), is the set of all subsets of $X$ : $2^X = \{ A : A \subseteq X \}.$

Remark.

Intuition: If $X$ has $n$ elements then $2^X$ has $2^n$ elements — one for each binary choice of "in/out" for each element of $X$ . This is why the notation $2^X$ is standard. For an infinite set $X$ , $2^X$ is strictly "larger" than $X$ (Cantor's theorem), which foreshadows the distinction between countable and uncountable sets.

Throughout this chapter we use $\subset$ and $\subseteq$ interchangeably to mean "is a subset of" (allowing equality). For a subset $A \subseteq X$ , the complement of $A$ in $X$ is $A^c = X \setminus A = \{ x \in X : x \notin A \}.$

Two identities we use constantly are de Morgan's laws: for any family $(A_\alpha)_{\alpha \in \mathcal{A}}$ of subsets of $X$ , $\left( \bigcup_{\alpha \in \mathcal{A}} A_\alpha \right)^c = \bigcap_{\alpha \in \mathcal{A}} A_\alpha^c, \qquad \left( \bigcap_{\alpha \in \mathcal{A}} A_\alpha \right)^c = \bigcup_{\alpha \in \mathcal{A}} A_\alpha^c.$

Remark.

Aside on sigma-algebras. A related but different structure on a set $X$ is a $\sigma$ -algebra: a collection $\mathcal{F} \subseteq 2^X$ closed under complements, countable unions, and containing $\emptyset$ and $X$ . Sigma-algebras are the foundation for measure theory. Topologies and $\sigma$ -algebras share some flavour (closure under certain set operations) but differ in crucial ways: topologies allow arbitrary unions but only finite intersections; $\sigma$ -algebras allow countable unions and countable intersections and require closure under complements. Do not conflate them.

Definition of a Topology

DefinitionTopology, Topological Space, Open Sets, Closed Sets

Let $X$ be a nonempty set. A collection $\mathcal{T} \subseteq 2^X$ is called a topology on $X$ if and only if:

$\emptyset \in \mathcal{T}$ and $X \in \mathcal{T}$ .
For every $\Omega_1, \Omega_2 \in \mathcal{T}$ , we have $\Omega_1 \cap \Omega_2 \in \mathcal{T}$ (closure under finite intersections).
For every family $(\Omega_\alpha)_{\alpha \in \mathcal{A}}$ with $\Omega_\alpha \in \mathcal{T}$ for all $\alpha$ , we have $\bigcup_{\alpha \in \mathcal{A}} \Omega_\alpha \in \mathcal{T}$ (closure under arbitrary unions).

The pair $(X, \mathcal{T})$ is called a topological space. Elements of $\mathcal{T}$ are called open sets. A subset $F \subseteq X$ is called a closed set if and only if $F^c = X \setminus F \in \mathcal{T}$ .

Remark.

Intuition: A topology is a choice of which subsets we declare to be "open." The three axioms encode what we expect open sets to do: both the ambient space and nothing at all are open; intersecting two open sets leaves an open set; taking the union of any (even uncountably many) open sets keeps us open. Note the asymmetry — intersections are only guaranteed to stay open when finitely many sets are involved. We will see examples where an infinite intersection of open sets fails to be open.

Remark.

Why these axioms? The axioms are the minimal conditions needed so that the following familiar notions — limit of a sequence, continuity of a function, closure of a set — can be defined without any reference to distance. Everything in the first third of this course flows from these three bullet points.

Closed Sets from the Topology

Closed sets are not a separate structure; they are entirely determined by the topology. The behavior of closed sets is dual to that of open sets, as the next proposition shows.

PropositionCharacterization of the Family of Closed Sets

Let $(X, \mathcal{T})$ be a topological space and let $\mathcal{F} = \{ F \subseteq X : F^c \in \mathcal{T} \}$ be the family of all closed sets. Then:

$\emptyset \in \mathcal{F}$ and $X \in \mathcal{F}$ .
For every $F_1, F_2 \in \mathcal{F}$ , we have $F_1 \cup F_2 \in \mathcal{F}$ .
For every family $(F_\alpha)_{\alpha \in \mathcal{A}}$ with $F_\alpha \in \mathcal{F}$ for all $\alpha$ , we have $\bigcap_{\alpha \in \mathcal{A}} F_\alpha \in \mathcal{F}$ .

Remark.

Intuition: The closed-set axioms mirror the open-set axioms with roles of union/intersection swapped. In particular, arbitrary intersections of closed sets are closed, but only finite unions of closed sets are guaranteed to be closed.

Clopen Sets

Remark.

In every topological space $(X, \mathcal{T})$ , the sets $\emptyset$ and $X$ are both open (by axiom 1) and closed (by the proposition above). Sets that are simultaneously open and closed are called clopen. Depending on the topology, there can be many clopen sets or only the trivial ones $\emptyset$ and $X$ .

Examples of Topologies

The following four examples are the central ones to have in mind.

ExampleThe Trivial (Indiscrete) Topology

For any nonempty set $X$ , the collection $\mathcal{T}_{\text{triv}} = \{ \emptyset, X \}$ is a topology on $X$ . All three axioms are trivially satisfied. This is the trivial (or indiscrete) topology — the fewest possible open sets.

In this topology, the only open sets are $\emptyset$ and $X$ , and the only closed sets are likewise $\emptyset$ and $X$ . Every pair of points is "topologically indistinguishable" in a sense we will make precise shortly (they cannot be separated by open sets).

ExampleThe Discrete Topology

For any nonempty set $X$ , the collection $\mathcal{T}_{\text{disc}} = 2^X$ of all subsets of $X$ is a topology on $X$ . The three axioms hold because every subset is declared open. This is the discrete topology — the most possible open sets.

In the discrete topology, every subset is both open and closed (every subset is clopen). In particular, every singleton $\{x\}$ is open.

ExampleThe Euclidean Topology on R to the n

Let $\mathbb{R}^n$ denote the set of $n$ -tuples of real numbers, equipped with the Euclidean norm $\|\boldsymbol{x}\| = \left( \sum_{j=1}^n x_j^2 \right)^{1/2}, \qquad \boldsymbol{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n.$ For $\boldsymbol{x} \in \mathbb{R}^n$ and $r > 0$ , define the open ball of radius $r$ centered at $\boldsymbol{x}$ : $B(\boldsymbol{x}, r) = \{ \boldsymbol{y} \in \mathbb{R}^n : \| \boldsymbol{y} - \boldsymbol{x} \| < r \}.$ Then define $\mathcal{E} = \{ \Omega \subseteq \mathbb{R}^n : \forall \boldsymbol{x} \in \Omega, \, \exists \rho > 0 \text{ such that } B(\boldsymbol{x}, \rho) \subseteq \Omega \}.$ The collection $\mathcal{E}$ is the Euclidean topology on $\mathbb{R}^n$ . This is exactly the usual definition of "open" from calculus: $\Omega$ is open if around every point in $\Omega$ there is some open ball entirely inside $\Omega$ . When $n = 1$ , we have $\|x\| = |x|$ and $B(x, r) = (x-r, x+r)$ .

TheoremThe Euclidean Topology is a Topology

The collection $\mathcal{E}$ defined above is a topology on $\mathbb{R}^n$ .

Remark.

Intuition: The Euclidean topology is what you were implicitly using in first-year calculus whenever you said "open interval" or "continuous function." The topological axioms above are the abstract distillation of exactly what makes proofs in calculus work.

ExampleOpen and Closed Sets in the Euclidean Topology on R

Consider $\mathbb{R}$ with the Euclidean topology $\mathcal{E}$ .

For $a < b$ , the interval $(a,b)$ is open. Given $x \in (a,b)$ , set $\rho = \min\{x - a, b - x\} > 0$ . Then $B(x, \rho) = (x - \rho, x + \rho) \subseteq (a,b)$ .
For every $a \in \mathbb{R}$ , the rays $(-\infty, a)$ and $(a, \infty)$ are open (similar argument).
For $a \le b$ , the interval $[a,b]$ is closed. Its complement $[a,b]^c = (-\infty, a) \cup (b, \infty)$ is a union of two open sets, hence open.
The rays $(-\infty, a]$ and $[a, \infty)$ are closed (by similar complement arguments).
The half-open interval $[a,b)$ with $a < b$ is neither open nor closed. It is not open because $a \in [a,b)$ and for every $\rho > 0$ , $B(a, \rho) = (a - \rho, a + \rho) \not\subseteq [a,b)$ . It is not closed because its complement $(-\infty, a) \cup [b, \infty)$ is not open: $b \in [b, \infty)$ but $B(b, \rho) \not\subseteq (-\infty, a) \cup [b, \infty)$ for any $\rho > 0$ .

ExampleThe Cofinite Topology

Let $X$ be any nonempty set. Define $\mathcal{T}_{\text{cofin}} = \{ \emptyset \} \cup \{ \Omega \subseteq X : X \setminus \Omega \text{ is finite} \}.$ One can verify that this is a topology: the empty set is in by definition; $X \setminus X = \emptyset$ is finite so $X$ is in; if $\Omega_1, \Omega_2$ have finite complements, then $(\Omega_1 \cap \Omega_2)^c = \Omega_1^c \cup \Omega_2^c$ is a union of two finite sets, hence finite; and an arbitrary union of sets with finite complements has a complement equal to the intersection of finite sets, which is finite. In the cofinite topology, the closed sets are exactly $X$ and the finite subsets of $X$ .

ExampleA Non-Hausdorff Topology on R

On $\mathbb{R}$ , define $\mathcal{T}_{\text{lsc}} = \{ (a, \infty) : a \in \mathbb{R} \} \cup \{ \emptyset, \mathbb{R} \}.$ This is a topology (the subscript "lsc" hints at its connection to lower-semicontinuous functions):

$\emptyset, \mathbb{R} \in \mathcal{T}_{\text{lsc}}$ by definition.
Intersections: $(a_1, \infty) \cap (a_2, \infty) = (\max\{a_1, a_2\}, \infty) \in \mathcal{T}_{\text{lsc}}$ , and intersections involving $\emptyset$ or $\mathbb{R}$ remain in the family.
Unions: a union of sets of the form $(a_\alpha, \infty)$ is either $\emptyset$ , $\mathbb{R}$ , or another ray $(a, \infty)$ where $a = \inf_\alpha a_\alpha$ .

This topology has far fewer open sets than the Euclidean topology. We will see that it is not Hausdorff — distinct points cannot be separated by disjoint open sets.

Closure Under Intersections: Finite vs. Arbitrary

The axioms of a topology allow finite intersections of open sets, but an infinite intersection of open sets may fail to be open. The following example in $(\mathbb{R}, \mathcal{E})$ illustrates this.

ExampleAn Infinite Intersection of Open Sets Can Fail to be Open

For each $k \in \mathbb{Z}_{>0}$ , the set $\left( -\tfrac{1}{k}, \tfrac{1}{k} \right)$ is open in $\mathbb{R}$ . Their intersection is $\bigcap_{k=1}^\infty \left( -\tfrac{1}{k}, \tfrac{1}{k} \right) = \{0\},$ which is a single point. In $(\mathbb{R}, \mathcal{E})$ , the singleton $\{0\}$ is not open — there is no $\rho > 0$ with $B(0, \rho) = (-\rho, \rho) \subseteq \{0\}$ . So the intersection of infinitely many open sets need not be open.

Similarly, an infinite union of closed sets need not be closed. Each closed ball $\overline{B}_{(k-1)/k}(\boldsymbol{x}) = \{ \boldsymbol{y} : \| \boldsymbol{y} - \boldsymbol{x} \| \le (k-1)/k \}$ is closed, but $\bigcup_{k=1}^\infty \overline{B}_{(k-1)/k}(\boldsymbol{x}) = B_1(\boldsymbol{x}),$ which is the open ball of radius 1 — not closed.

Neighbourhoods

DefinitionNeighbourhood

Let $(X, \mathcal{T})$ be a topological space and let $x \in X$ . A subset $V \subseteq X$ is called a neighbourhood of $x$ (in the topology $\mathcal{T}$ ) if and only if there exists $\Omega \in \mathcal{T}$ such that $x \in \Omega \subseteq V.$ Given $x \in X$ , we denote by $\mathcal{U}_x$ the family of all neighbourhoods of $x$ in the topology $\mathcal{T}$ .

Remark.

Intuition: A neighbourhood of $x$ is any set "fat enough around $x$ " to contain an open set through $x$ . Neighbourhoods need not themselves be open — for instance, in $\mathbb{R}$ , the closed interval $[-1, 1]$ is a neighbourhood of 0 because it contains the open interval $(-1, 1) \ni 0$ .

The family $\mathcal{U}_x$ has four basic properties:

$V \in \mathcal{U}_x \implies x \in V$ .
If $V \in \mathcal{U}_x$ and $W \supseteq V$ , then $W \in \mathcal{U}_x$ .
$V_1, V_2 \in \mathcal{U}_x \implies V_1 \cap V_2 \in \mathcal{U}_x$ .
For every $V \in \mathcal{U}_x$ , there exists $W \in \mathcal{U}_x$ such that $V \in \mathcal{U}_y$ for every $y \in W$ .

Properties (1)–(3) are immediate from the definition. Property (4) says: every neighbourhood of $x$ is a neighbourhood of all points in some smaller neighbourhood. To see this, let $V \in \mathcal{U}_x$ ; by definition there exists $W \in \mathcal{T}$ with $x \in W \subseteq V$ . Then $W \in \mathcal{U}_x$ , and for every $y \in W$ , $W$ is an open set containing $y$ with $W \subseteq V$ , so $V \in \mathcal{U}_y$ .

PropositionOpenness via Neighbourhoods

Let $(X, \mathcal{T})$ be a topological space and let $\Omega \subseteq X$ . Then $\Omega \in \mathcal{T} \iff \Omega \in \mathcal{U}_x \text{ for every } x \in \Omega.$ In words: a set is open if and only if it is a neighbourhood of each of its points.

Remark.

Hausdorff's viewpoint. Specifying, for each $x \in X$ , a family $\mathcal{U}_x$ of subsets satisfying properties (1)–(4), determines a unique topology $\mathcal{T}$ on $X$ such that $\mathcal{U}_x$ is the system of neighbourhoods of $x$ in the topology $\mathcal{T}$ . This was the approach Felix Hausdorff (1868–1942) took to define topological spaces in 1914. Modern treatments favour the open-set axioms, but the two approaches are equivalent.

Hausdorff Spaces

DefinitionHausdorff Space

A topological space $(X, \mathcal{T})$ is called a Hausdorff space (or $T_2$ -space) if and only if for every $x, y \in X$ with $x \ne y$ , there exist $V \in \mathcal{U}_x$ and $W \in \mathcal{U}_y$ such that $V \cap W = \emptyset$ .

Remark.

Intuition: In a Hausdorff space, distinct points can be "separated" by disjoint neighbourhoods. This is the minimum amount of separation we generally want to do analysis: without it, limits of sequences might fail to be unique, and points might be topologically indistinguishable. Almost every space arising in practice (every metric space, in particular) is Hausdorff.

ExampleA Non-Hausdorff Topology

Consider $(\mathbb{R}, \mathcal{T}_{\text{lsc}})$ from earlier. This is not Hausdorff: let $x, y \in \mathbb{R}$ with $x < y$ . Every nonempty proper open set in $\mathcal{T}_{\text{lsc}}$ is of the form $(a, \infty)$ . Every neighbourhood of $x$ must contain some $(a, \infty)$ with $a < x$ , and therefore contains all of $(x, \infty)$ , which in turn contains $y$ . So every neighbourhood of $x$ and every neighbourhood of $y$ both contain the interval $(y, \infty)$ and cannot be disjoint.

The key intuition behind the failure: the topology $\mathcal{T}_{\text{lsc}}$ has many fewer open sets than the Euclidean topology. This implies fewer neighbourhoods at our disposal when attempting to separate points; the separation attempt fails.

Similarly, if $X$ has at least two elements, the trivial topology $(X, \{\emptyset, X\})$ is not Hausdorff, since the only neighbourhood of any point is $X$ itself — so two distinct points can never be placed in disjoint neighbourhoods.

Accumulation Points (Cluster Points) and the Derived Set

Before turning to limits and continuity, we introduce a key notion — points that are "approached" by a set, which formalizes the idea of a set piling up near a point.

DefinitionAccumulation Point, Derived Set

Let $(X, \mathcal{T})$ be a topological space, let $A \subseteq X$ , and let $x_0 \in X$ . We say $x_0$ is an accumulation point (or cluster point, or limit point) of $A$ if and only if $(A \setminus \{x_0\}) \cap V \ne \emptyset \qquad \text{for every } V \in \mathcal{U}_{x_0}.$ The set of all accumulation points of $A$ is called the derived set of $A$ and is denoted $A'$ .

This notion is due to Georg Cantor (1872).

Remark.

Intuition: $x_0$ is an accumulation point of $A$ if every neighbourhood of $x_0$ meets $A$ in some point other than $x_0$ itself. So $A$ "piles up" near $x_0$ . Note carefully: this differs from saying "every neighbourhood meets $A$ ." We forbid the meeting from being only at $x_0$ — there must be another point of $A$ nearby.

In a metric space $(X, d)$ , equipped with the metric topology, the definition can be rewritten in terms of open balls: $x_0 \in A' \iff (A \setminus \{x_0\}) \cap B(x_0, \rho) \ne \emptyset \quad \text{for every } \rho > 0.$

Remark.

Alternative characterization in Hausdorff spaces. Another common definition of "accumulation point" (often seen first in $\mathbb{R}^n$ ) is: $x_0 \in A' \iff A \cap V \text{ is infinite for every } V \in \mathcal{U}_{x_0}.$ This equivalence holds when $(X, \mathcal{T})$ is Hausdorff (and so, in particular, in every metric space). It need not hold in general non-Hausdorff spaces.

Limits of Functions Between Topological Spaces

With accumulation points in hand, we can state the topological definition of limit.

DefinitionLimit of a Function

Let $(X, \mathcal{T})$ and $(Y, \mathcal{S})$ be topological spaces. Let $A \subseteq X$ and $f : A \to Y$ be a function. Let $a \in A'$ (an accumulation point of $A$ ) and $\lambda \in Y$ . We say that $\lambda$ is a limit of $f$ as $x$ tends to $a$ , and write $\lim_{x \to a} f(x) = \lambda$ , if and only if $\forall V \in \mathcal{U}_\lambda, \, \exists W \in \mathcal{U}_a \text{ such that } f(x) \in V \text{ for every } x \in (A \setminus \{a\}) \cap W.$ To stress the dependence on the topologies $\mathcal{T}$ and $\mathcal{S}$ , we may write $f(x) \xrightarrow{\mathcal{S}} \lambda$ as $x \xrightarrow{\mathcal{T}} a$ .

Remark.

Intuition: The statement $\lim_{x \to a} f(x) = \lambda$ says: however tight a neighbourhood $V$ of $\lambda$ we pick in the target, we can find a neighbourhood $W$ of $a$ in the source so small that $f$ maps every point of $A$ different from $a$ and inside $W$ into $V$ . Requiring $a \in A'$ ensures that such "points near $a$ but not equal to $a$ " actually exist in $A$ .

ExampleLimits Need Not Be Unique

In a non-Hausdorff space, limits can fail to be unique. Consider $(\mathbb{R}, \mathcal{T}_{\text{lsc}})$ and suppose $f(x) \to \lambda$ in this topology. We claim that $f(x) \to \mu$ for every $\mu \le \lambda$ .

Indeed, let $V_1 \in \mathcal{U}_\mu$ in $\mathcal{T}_{\text{lsc}}$ . By definition there is $\alpha \in \mathbb{R}$ with $\mu \in (\alpha, \infty) \subseteq V_1$ . Since $\alpha < \mu \le \lambda$ , we have $\lambda \in (\alpha, \infty) \subseteq V_1$ , so $V_1$ is also a neighbourhood of $\lambda$ . Hence by convergence to $\lambda$ there exists $W_1 \in \mathcal{U}_a$ with $f(x) \in V_1$ for every $x \in (A \setminus \{a\}) \cap W_1$ . This is the condition for convergence to $\mu$ . So the same function simultaneously converges to $\lambda$ and to every $\mu \le \lambda$ .

TheoremUniqueness of Limits in Hausdorff Target

Let $(X, \mathcal{T})$ and $(Y, \mathcal{S})$ be topological spaces with $(Y, \mathcal{S})$ Hausdorff. Let $A \subseteq X$ , $f : A \to Y$ , and $a \in A'$ . If $\lim_{x \to a} f(x) = \lambda$ and $\lim_{x \to a} f(x) = \mu$ , then $\lambda = \mu$ .

Remark.

Consequence for metric spaces. Since every metric space is Hausdorff, limits of functions taking values in a metric space (with the metric topology) are unique whenever they exist. In particular, limits of sequences in metric spaces are unique.

The Extended Real Line

The topological framework is broad enough to handle limits like $\lim_{x \to +\infty} e^{-x} = 0$ , $\lim_{x \to 0} 1/x^2 = +\infty$ , and $\lim_{x \to 2} \log|x - 2| = -\infty$ — all within the same neighbourhood-based definition of limit. To do this, we extend $\mathbb{R}$ with two new points and define a topology that makes the familiar notion of "convergence at infinity" a special case of the topological definition.

DefinitionExtended Real Line

Set $\overline{\mathbb{R}} = \mathbb{R} \cup \{+\infty, -\infty\}$ . For $r > 0$ , define $B(+\infty, r) = \left( \tfrac{1}{r}, +\infty \right) \cup \{+\infty\}, \qquad B(-\infty, r) = \left( -\infty, -\tfrac{1}{r} \right) \cup \{-\infty\}.$ These play the role of "balls centered at $\pm\infty$ ." Then define $\overline{\mathcal{E}} = \{ \Omega \subseteq \overline{\mathbb{R}} : \forall x \in \Omega, \, \exists \rho > 0 \text{ with } B(x, \rho) \subseteq \Omega \},$ where for $x \in \mathbb{R}$ the ball $B(x, \rho)$ is the ordinary Euclidean open ball $(x - \rho, x + \rho)$ . The collection $\overline{\mathcal{E}}$ is a topology (exercise); the topological space $(\overline{\mathbb{R}}, \overline{\mathcal{E}})$ is called the extended real line.

Remark.

Intuition: A neighbourhood of $+\infty$ is a set containing some tail $(M, \infty) \cup \{+\infty\}$ ; smaller $r$ corresponds to larger $M = 1/r$ , i.e., more restrictive tails. This is exactly the "arbitrarily large" condition we use when saying $x \to +\infty$ .

On the extended real line, $+\infty$ is an accumulation point of a set $A \subseteq \mathbb{R} \subseteq \overline{\mathbb{R}}$ if and only if $\sup A = +\infty$ ; similarly $-\infty \in A'$ iff $\inf A = -\infty$ . For a function $f : A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ and $\sup A = +\infty$ , treating domain and codomain as subsets of $\overline{\mathbb{R}}$ gives $\lim_{x \to +\infty} f(x) = \lambda \in \mathbb{R} \iff \forall \varepsilon > 0 \, \exists M > 0 \text{ such that } f(x) \in (\lambda - \varepsilon, \lambda + \varepsilon) \, \forall x \in (M, \infty) \cap A.$ A special case with $A = \mathbb{N}$ and $f(n) = a_n$ recovers the familiar definition of the limit of a sequence. Similarly, $\lim_{x \to a} f(x) = +\infty \iff \forall M > 0 \, \exists \delta > 0 \text{ such that } f(x) > M \, \forall x \in (A \setminus \{a\}) \cap (a - \delta, a + \delta),$ and analogous statements for all other combinations of finite or infinite $a$ and $\lambda$ .

Remark.

Intuition: Adding $\pm\infty$ as genuine points and choosing their neighbourhoods to be "tails" makes the ordinary topological definition of limit coincide with every flavour of limit seen in first-year calculus — limits at infinity, limits equal to infinity, and limits of sequences. Everything becomes the same definition.

Bases for a Topology

Sometimes it is inconvenient to describe a topology by listing every open set; instead, we describe a smaller collection from which all open sets can be built by unions.

DefinitionBasis for a Topology

Let $X$ be a nonempty set. A collection $\mathcal{B} \subseteq 2^X$ is a basis for a topology on $X$ if:

For every $x \in X$ , there exists $B \in \mathcal{B}$ with $x \in B$ (the basis covers $X$ ).
For every $B_1, B_2 \in \mathcal{B}$ and every $x \in B_1 \cap B_2$ , there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$ .

The topology generated by $\mathcal{B}$ is $\mathcal{T}_\mathcal{B} = \{ \Omega \subseteq X : \forall x \in \Omega, \, \exists B \in \mathcal{B} \text{ with } x \in B \subseteq \Omega \}.$ Equivalently, $\Omega \in \mathcal{T}_\mathcal{B}$ if and only if $\Omega$ is a union of elements of $\mathcal{B}$ .

Remark.

Intuition: A basis is a "generating set" for a topology. The Euclidean topology on $\mathbb{R}^n$ is generated by the collection of all open balls $\{ B(\boldsymbol{x}, r) : \boldsymbol{x} \in \mathbb{R}^n, \, r > 0 \}$ — every open set is a union of open balls, and this is often an easier way to describe the topology than specifying every open set directly. The axioms on $\mathcal{B}$ guarantee that the generated $\mathcal{T}_\mathcal{B}$ is genuinely a topology.

PropositionOpen Balls Form a Basis for the Euclidean Topology

The collection $\mathcal{B} = \{ B(\boldsymbol{x}, r) : \boldsymbol{x} \in \mathbb{R}^n, \, r > 0 \}$ is a basis, and the topology it generates is the Euclidean topology $\mathcal{E}$ .

The Subspace (Relative) Topology

Given a topological space, any subset inherits a natural topology.

DefinitionSubspace Topology

Let $(X, \mathcal{T})$ be a topological space and let $A \subseteq X$ . The subspace topology (or relative topology, or induced topology) on $A$ is $\mathcal{T}_A = \{ A \cap \Omega : \Omega \in \mathcal{T} \}.$ Sets in $\mathcal{T}_A$ are called open relative to $A$ (or open in $A$ ).

PropositionThe Subspace Topology is a Topology

$\mathcal{T}_A$ is a topology on $A$ .

Remark.

Intuition: The subspace topology is exactly the "restriction" of the topology on $X$ to $A$ : a set is open in $A$ iff it is the intersection of $A$ with an open set from $X$ . Concretely, $[0, \tfrac12)$ is open in the subspace $[0, 1]$ of $\mathbb{R}$ (it's the intersection of $[0,1]$ with the open set $(-1, \tfrac12)$ ), even though it's not open in $\mathbb{R}$ . The closed sets in $\mathcal{T}_A$ are exactly the sets of the form $A \cap F$ where $F$ is closed in $X$ .

Continuity Between Topological Spaces

We close the chapter with one of the central definitions of topology — continuity — formulated purely in terms of open sets.

DefinitionContinuity at a Point

Let $(X, \mathcal{T})$ and $(Y, \mathcal{S})$ be topological spaces, let $A \subseteq X$ , let $a \in A$ , and let $f : A \to Y$ be a function. We say $f$ is continuous at $a$ if and only if $\forall V \in \mathcal{U}_{f(a)}, \, \exists W \in \mathcal{U}_a \text{ such that } f(A \cap W) \subseteq V.$ We say $f$ is continuous if it is continuous at every $a \in A$ .

Remark.

Intuition: The definition says: given any target neighbourhood $V$ of the value $f(a)$ , we can find a source neighbourhood $W$ of $a$ so small that $f$ maps $A \cap W$ entirely inside $V$ . This is the topological rendering of the $\varepsilon$ - $\delta$ definition — the $\varepsilon$ -ball around $f(a)$ becomes "any neighbourhood $V$ " and the $\delta$ -ball around $a$ becomes "some neighbourhood $W$ ."

Remark.

Isolated points. If $a \in A$ is isolated (i.e., $\{a\}$ is open in the subspace topology $\mathcal{T}_A$ ), then $f$ is automatically continuous at $a$ : take $W$ to be the open set witnessing that $\{a\}$ is open in $A$ ; then $A \cap W = \{a\}$ , so $f(A \cap W) = \{f(a)\} \subseteq V$ for every $V \in \mathcal{U}_{f(a)}$ . So every function is continuous at every isolated point of its domain.

TheoremContinuity via Preimages of Open Sets

Let $(X, \mathcal{T})$ and $(Y, \mathcal{S})$ be topological spaces, let $A \subseteq X$ , and let $f : A \to Y$ . Then $f$ is continuous if and only if for every $O \in \mathcal{S}$ there exists $\Omega \in \mathcal{T}$ such that $f^{-1}(O) = A \cap \Omega.$ In words: $f$ is continuous if and only if the preimage of every open set in $Y$ is relatively open in $A$ .

Remark.

Intuition: The preimage characterization is usually the most convenient definition to work with. Observe that the condition is " $f^{-1}(O)$ is relatively open in $A$ ," not " $f^{-1}(O)$ is open in $X$ " — this is because the domain of $f$ is $A$ , not all of $X$ . When $A = X$ , the condition simplifies to " $f^{-1}(O) \in \mathcal{T}$ for every $O \in \mathcal{S}$ ."

CorollaryComposition of Continuous Functions is Continuous

Let $(X, \mathcal{T})$ , $(Y, \mathcal{S})$ , $(Z, \mathcal{R})$ be topological spaces and let $f : X \to Y$ and $g : Y \to Z$ be continuous. Then $g \circ f : X \to Z$ is continuous.

Looking Ahead

We have developed the abstract framework of topology. The next step is to specialize: most topological spaces encountered in analysis come equipped with a distance function (a metric), which automatically generates a topology through open balls. Metric spaces are the subject of the next chapter, and they form the arena for most of the rest of real analysis — sequences, completeness, compactness, and the basic theorems of calculus all live most naturally there.