13.1 Topological Spaces, Open/Closed Sets, and Bases

13.1 Topological Spaces, Open/Closed Sets, and Bases

Point-set topology abstracts the core notions of distance, convergence, and continuity onto general sets without a metric. On the GRE Mathematics Subject Test, questions test axiomatic definitions, finer versus coarser topologies, operators (interior, closure, boundary), and countability axioms.

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Topological Spaces and Axiomatic Foundations

A topological space is a pair $(X, \mathcal{T})$, where $X$ is a set and $\mathcal{T} \subseteq \mathcal{P}(X)$ is a collection of subsets called open sets satisfying:

1. $\emptyset \in \mathcal{T}$ and $X \in \mathcal{T}$.
2. If ${U_\alpha}{\alpha \in I} \subseteq \mathcal{T}$, then $\bigcup{\alpha \in I} U_\alpha \in \mathcal{T}$ (arbitrary unions are open).
3. If $U_1, \dots, U_n \in \mathcal{T}$, then $\bigcap_{k=1}^n U_k \in \mathcal{T}$ (finite intersections are open).

Canonical Topologies on a Set $X$

• Discrete Topology: $\mathcal{T}_{\text{disc}} = \mathcal{P}(X)$. Every subset is open; finest topology on $X$.
• Indiscrete (Trivial) Topology: $\mathcal{T}_{\text{triv}} = {\emptyset, X}$. Coarsest topology on $X$.
• Standard (Euclidean) Topology on $\mathbb{R}$: Generated by all open intervals $(a, b)$.
• Lower Limit Topology ($\mathbb{R}_\ell$, Sorgenfrey Line): Generated by all half-open intervals $[a, b)$.
• Cofinite (Finite Complement) Topology: $\mathcal{T}_{\text{cof}} = {U \subseteq X : X \setminus U \text{ is finite}} \cup {\emptyset}$.
• Cocountable Topology: $\mathcal{T}_{\text{coc}} = {U \subseteq X : X \setminus U \text{ is countable}} \cup {\emptyset}$.

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Comparison of Topologies: Finer vs. Coarser

Given two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ on the same set $X$:

• If $\mathcal{T}_1 \subseteq \mathcal{T}_2$, then $\mathcal{T}_2$ is finer (stronger) than $\mathcal{T}_1$, and $\mathcal{T}_1$ is coarser (weaker) than $\mathcal{T}_2$.
• On $\mathbb{R}$, two valid strict chains are: $\mathcal{T}_{\text{triv}} \subsetneq \mathcal{T}_{\text{cof}} \subsetneq \mathcal{T}_{\text{coc}} \subsetneq \mathcal{T}_{\text{disc}},$ $\mathcal{T}_{\text{triv}} \subsetneq \mathcal{T}_{\text{std}} \subsetneq \mathcal{T}_\ell \subsetneq \mathcal{T}_{\text{disc}}.$
• The cocountable and standard Euclidean topologies are incomparable. The set $\mathbb{R}\setminus\mathbb{Q}$ is cocountable-open but not Euclidean-open, while $(0,1)$ is Euclidean-open but not cocountable-open because its complement is uncountable.
• Why $\mathbb{R}\ell$ is Strictly Finer than $\mathcal{T}{\text{std}}$: Any open interval $(a, b) = \bigcup_{n=1}^\infty [a + 1/n, b)$ is a union of lower-limit basis elements, so $(a, b) \in \mathcal{T}\ell$. However, $[a, b) \notin \mathcal{T}{\text{std}}$ because no Euclidean open ball centered at $a$ lies within $[a, b)$.

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Closed Sets, Clopen Sets, and Topological Operations

A set $F \subseteq X$ is closed if its complement $X \setminus F$ is open. A set is clopen if it is both open and closed. In $\mathbb{R}\ell$, every basis interval $[a, b)$ is clopen because its complement $(-\infty, a) \cup [b, \infty) = \bigcup{n=1}^\infty [a-n, a) \cup \bigcup_{k=1}^\infty [b, b+k)$ is open.

Core Topological Operations on $A \subseteq X$

• Interior: $\operatorname{int}(A) = A^\circ = \bigcup {U \in \mathcal{T} : U \subseteq A}$ is the largest open set contained in $A$. Pointwise, $x \in \operatorname{int}(A) \iff \exists U \in \mathcal{T}$ with $x \in U \subseteq A$.
• Closure: $\operatorname{cl}(A) = \overline{A} = \bigcap {F \text{ closed} : A \subseteq F}$ is the smallest closed set containing $A$. Pointwise, $x \in \overline{A} \iff \forall U \in \mathcal{T}$ containing $x$, $U \cap A \neq \emptyset$.
• Boundary: $\partial A = \overline{A} \setminus \operatorname{int}(A) = \overline{A} \cap \overline{X \setminus A}$, always closed.
• Limit Points & Derived Set: $x \in X$ is a limit point of $A$ if every neighborhood $U$ of $x$ satisfies $(U \setminus {x}) \cap A \neq \emptyset$. The set of all limit points is $A'$, and $\overline{A} = A \cup A'$.
• Isolated Points & Dense Sets: A point $x \in A$ is isolated if ${x} = U \cap A$ for some open $U$. A set $D \subseteq X$ is dense if $\overline{D} = X$.

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Bases, Subbases, and Countability Axioms

Bases and Subbases

• Basis $\mathcal{B}$: A collection of open sets such that $\bigcup_{B \in \mathcal{B}} B = X$, and if $x \in B_1 \cap B_2$, there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$. The topology consists of all unions of elements of $\mathcal{B}$.
• Subbasis $\mathcal{S}$: A collection whose union is $X$; its finite intersections form a basis for $\mathcal{T}$.

Countability Axioms

• First-Countable: Every point $x \in X$ has a countable local base $\mathcal{B}x$. All metric spaces are first-countable (take $B{1/n}(x)$).
• Second-Countable: $X$ has a countable basis for its entire topology.
• Separable: $X$ contains a countable dense subset (e.g., $\mathbb{Q} \subset \mathbb{R}$).
• The Metric Equivalence: For metric spaces, Second-Countable $\iff$ Separable $\iff$ Lindelöf (every open cover has a countable subcover).
• The Sorgenfrey Counterexample: $\mathbb{R}_\ell$ is first-countable and separable ($\mathbb{Q}$ is dense), but not second-countable.

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Comparison Table of Topologies on $\mathbb{R}$

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Step-by-Step Worked Problems

Problem 1: Operators in the Cofinite Topology

Let $X = \mathbb{R}$ have the cofinite topology $\mathcal{T}_{\text{cof}}$. Find $\operatorname{int}(A)$ and $\overline{A}$ for $A = (0, 1) \cup {2}$.

Solution:

1. Non-empty open sets have finite complements; closed sets are finite subsets of $\mathbb{R}$ plus $\mathbb{R}$.
2. If $U \subseteq A$ is open and non-empty, $\mathbb{R} \setminus U$ is finite, so $U$ must omit only finitely many points. Because $A$ omits an uncountable set, no non-empty open set fits inside $A$. Thus, $\operatorname{int}(A) = \emptyset$.
3. Closed sets containing $A$ must contain an infinite set, so the only closed set containing $A$ is $\mathbb{R}$. Therefore, $\overline{A} = \mathbb{R}$.

Problem 2: Rational Endpoints and Topology Comparison

Determine whether $\mathcal{B} = {[a, b) : a < b, , a, b \in \mathbb{Q}}$ is a basis for a topology on $\mathbb{R}$, and compare it to $\mathbb{R}_\ell$.

Solution:

1. For any $x$, choose rationals $q_1 \le x < q_2$, so $x \in [q_1, q_2)$. The intersection of $[a_1, b_1)$ and $[a_2, b_2)$ is $[\max(a_1, a_2), \min(b_1, b_2))$, which is in $\mathcal{B}$ or empty. Thus $\mathcal{B}$ is a basis.
2. For irrational $\alpha$, $[\alpha, \beta)$ is open in $\mathbb{R}\ell$ but cannot be a union of intervals with rational left endpoints. Hence the topology generated by $\mathcal{B}$ is strictly coarser than $\mathbb{R}\ell$.

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GRE Exam Traps & Pitfalls

Trap 1: The Binary Door Fallacy Sets need not be either open or closed: in standard $\mathbb{R}$, $(0, 1]$ is neither; in discrete spaces, every set is clopen.

Trap 2: Cofinite Accumulation Points In the cofinite topology on an infinite set $X$, every infinite set $A$ is dense ($\overline{A} = X$) and has derived set $A' = X$.

Trap 3: Separable vs. Second-Countable Second-countable always implies separable, but the converse is false in general topology: the Sorgenfrey line $\mathbb{R}_\ell$ is separable but not second-countable.