# Book:Lynn Arthur Steen/Counterexamples in Topology/Second Edition/Errata

## Errata for 1978: Lynn Arthur Steen: *Counterexamples in Topology* (2nd ed.)

### Union of Exteriors contains Exterior of Intersection

Part $\text I$: Basic Definitions: Section $1.$ General Introduction: Closures and Interiors

*The exterior of the union of sets is always contained in the intersection of the exteriors, and similarly, the exterior of the intersection is contained in the union of the exteriors; equality holds only for finite unions and intersections.*

### Compactness Properties Preserved under Continuous Mapping

Part $\text I$: Basic Definitions: Section $3$. Compactness: Invariance Properties

*To be precise, the properties of compactness, $\sigma$-compactness, countable compactness, sequential compactness, Lindelöf, and separability are preserved under continuous maps ... [Weak] local compactness, and first and second countability are preserved under open continuous maps, but not just under continuous maps ...*

### Equivalence of Metrics is not Defined

Part $\text I$: Basic Definitions: Section $5$. Metric Spaces: Complete Metric Spaces

- The concept of
**equivalence of metrics**is not defined, although the concept is mentioned and used in the context of complete metric spaces.

### Definition of Uniformity

Part $\text I$: Basic Definitions: Section $5$. Metric Spaces: Uniformities

*The quasiuniformity $\UU$ is a***uniformity**if the following additional condition is satisfied:*$\text U 5$: If $u \in \UU$, then $u^{-1} \in \UU$ where $u^{-1} = \set {\tuple {y, x}: \tuple {x, y} \in \UU}$.*

### Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space

Part $\text {II}$: Counterexamples: $9 \text { - } 10$: Infinite Particular Point Topology: Item $1$

*The sequences $\set {a_i}$ which converge are those for which the $a_i \ne p$ are equal for all but a finite number of indices. The only accumulation points for sequences are the points $b_j$ that the $a_i$ equal for infinitely many indices.*

### Non-Trivial Particular Point Topology is not $T_4$

Part $\text {II}$: Counterexamples: $8 \text { - } 10$: Particular Point Topology: Item $4$

*Every particular point topology is $T_0$, but since there are no disjoint open sets, none of the higher separation axioms are satisfied unless $X$ has only one point.*

### Particular Point Space is not Weakly Countably Compact

Part $\text {II}$: Counterexamples: $8 \text { - } 10$: Particular Point Topology: Item $12$

*[A particular point space] $X$ is not weakly countably compact since any set which does not contain $p$ has no limit points.*

### Sets in Modified Fort Space are Separated

Part $\text{II}$: Counterexamples: $27$: Modified Fort Space: Item $4$

### Kuratowski's Closure-Complement Problem

Part $\text {II}$: Counterexamples: $32$: Special Subsets of the Real Line: Item $9$: Figure $12$

Steen and Seebach present a more complicated $14$-set than is necessary to demonstrate the theorem:

- $A := \set {\tfrac 1 n: n \in \Z_{>0} } \cup \openint 2 3 \cup \openint 3 4 \cup \set {4 \tfrac 1 2} \cup \closedint 5 6 \cup \paren {\hointr 7 8 \cap \Q}$

They present Figure $12$ to illustrate the various generated subsets graphically:

The following mistakes can be identified in the above diagram:

- $(1): \quad$ The set $A$ as presented expresses the interval of rationals as closed, whereas it is in fact half open.

- $(2): \quad$ The sets are all presented as subsets of $\R_{\ge 0}$, while this is not stated in the text.

- $(3): \quad$ $0$ is erroneously excluded from $A^{\prime}$.

### Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected

Part $\text{II}$: Counterexamples: $33$: Special Subsets of the Plane: Item $2$

*Let $A$ be the subset of $\R^2 = \R \times \R$ consisting of all point with at least one irrational coordinate, and let $A$ have the induced topology. $A$ is arc-connected since a point $\tuple {x_1, y_1}$ with two irrational coordinates may be joined by an arc to any point $\tuple {a, b} \in A$ either $a$ or $b$ is irrational, say $a$. Then the union of the lines $x = a, y = y_1$ is an arc-connected subset of $X$ connecting $\tuple {x_1, y_1}$ to $\tuple {a, b}$. ...*

### Linearly Ordered Space is $T_5$

Part $\text {II}$: Counterexamples: $39$. Order Topology: Item $6$

*For each $\gamma$, select and fix some point $k_\gamma \in C_\gamma$. Then whenever $A_\alpha \cap \overline S_\alpha \ne \O$, there exists a unique ${k_\alpha}^+ \in {C_\alpha}^+$, the immediate successor of $A_\alpha$ ... otherwise, if $A_\alpha \cap {\overline S_\alpha}^\alpha = \O$, let $I_\alpha = \O$. ...*

### Telophase Topology

Part $\text {II}$: Counterexamples: $73$. Telophase Topology

*Let $\struct {X, \tau}$ be the topological space formed by adding to the ordinary closed unit topology $\sqbrk {0, 1}$ another right end point, say $1^*$, with the sets $\paren {\alpha, 1} \cup \set {1^*}$ as a local neighborhood basis.*

### Bibliography: Appert

Bibliography

- [10] Appert, Q.
*Propriétés des Espaces Abstraits les Plus Généraux*. Actual. Sci. Ind. No. 146, Herman, 1934