\(\DeclareMathOperator{\Fr}{Fr} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\mysharp}{\#} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \renewcommand{\Re}{\mathop{\textrm{Re}}} \renewcommand{\Im}{\mathop{\textrm{Im}}} \)

Comments on the book Elementary Topology

Here are some corrections and amplifications for the second edition of Elementary Topology by Michael C. Gemignani, used as a textbook in Math 436 during the Spring 2014 semester.

Remarks on language

  • The author refers to the reader with the masculine pronoun “he” in the generic sense (meaning “he or she”), an old-fashioned convention that has gone out of style since the book was written almost half a century ago.
  • The author frequently uses the word “any,” an ambiguous term that can mean either “some” or “every.” In the question, “Can anybody solve Exercise 3?”, the meaning of “any” is “some.” In the statement, “I can solve any exercise in Chapter 2,” the meaning of “any” is “every.” In ordinary English prose, the meaning of “any” usually is clear from context, but in mathematical prose, confusion can arise.
    In the terminology of Math 220, the word “some” corresponds to the existential quantifier \(\exists\), and the word “every” corresponds to the universal quantifier \(\forall\). When in doubt concerning the meaning of “any,” try replacing the word with the expression “an arbitrary.” Thus “if \(X\) is any metric space” translates to “if \(X\) is an arbitrary metric space,” so “any” means “some”; but “any bounded set of real numbers has a least upper bound” translates to “an arbitrary bounded set of real numbers has a least upper bound,” and now “any” means “every.”
  • The author calls a function “one-one” if distinct points of the domain map to distinct points of the range. More common terminology is “one-to-one” or “injective.” The author uses the word “onto” in the standard sense (for which “surjective” is a synonym), meaning that every point of the target space is actually in the image.

Remarks on notation

  • Some writers distinguish between the symbols \(\subseteq\) and \(\subset\), the first meaning subset and the second meaning proper subset. But this book uses only the symbol \(\subset\), which denotes subset in the broad sense (not necessarily a proper subset).
  • The author denotes the empty set by a symbol that looks much like the Greek letter \(\phi\) (phi). The modern symbol \(\varnothing\) for the empty set is less likely to be confused with the Greek letter.
  • The author denotes set difference by \(S\) — \(T\) with a horizontal dash. Nowadays a slanted line is commonly used: namely, \(S\setminus T\).
  • The “blackboard boldface” font came into common use after the book was written (although there are instances in print as early as 1965). Thus the author denotes the set of positive integers by \(N\), rather than by the modern symbol \(\N\), and the coordinate plane by \(R^2\) rather than by \(\R^2\). Be aware that in Section 4.5, the author uses the symbol \(R\) to denote an equivalence relation, although this symbol denotes the real numbers elsewhere in the book.
  • The author denotes the restriction of a function \(f\) to a set \(W\) by \(f\mid W\). Nowadays the usual notation is \(f\big|_W\) with the set written as a subscript.
  • A topological space consists of a set \(X\) together with a collection \(\tau\) of open sets. The author uses the notation \(X\), \(\tau\). Modern authors usually delimit such an ordered pair with parentheses: namely, \((X,\tau)\).
  • The author uses a prime symbol in the standard mathematical sense: thus \(x\) and \(x'\) are two objects of the same kind, possibly but not necessarily related to each other. But in Section 3.5, the prime symbol has a specialized meaning, indicating the derived set defined at the top of page 56. Sometimes the two different conventions appear on the same page (on page 69, for instance).
  • The author denotes the Cartesian product by \({\large\mathsf{X}}_I \, S_i\). A more common notation is \(\prod_I S_i\).

Other comments

Page 26, Definition 5
The author states that “capital \(N\) will be used almost exclusively in this text to denote the set of positive integers,” yet in the same paragraph, the notation \(N(y,p)\) refers to a neighborhood. Apparently what the author means is that the letter \(N\) denotes the set of positive integers when the letter is used as an isolated symbol.
Page 29, Exercise 5
Although Chapter 2 is about metric spaces, this exercise is outside the realm of metric spaces, for the indicated collections of open sets do not arise from any metric. The exercise is thus a preview of the notion of topological space to be studied in Chapter 3. In particular, see Example 4 on page 41.
Page 38, Exercise 5
The statement of the exercise is incorrect. What is true is that the distance from the set \(\{s_n\}\) to the point \(y\) is equal to zero if and only if some subsequence converges to \(y\).
Page 43, Exercise 4
This problem is contained in Exercise 6 on page 55.
Page 55, Exercise 4
The second part of the problem is misleading, for \(D''\) is not a metric (indeed, the triangle inequality does not hold). The problem would be more sensible if \(D''(x,y)\) were defined to be \(|x-y|^{1/2}\).
Page 63, Exercise 7
The equation \(O^n=0\) should say instead \(O^n=O\). The numeral \(0\) should be the letter \(O\).
Page 65, Example 2
The wording “all but at most finitely many” is confusing. Deleting the words “at most” would not essentially change the meaning. The author is trying to describe the cofinite topology, in which a nonempty set is open precisely when the complement is a finite set (possibly the empty set). But the author's description is deficient, since the empty set must be open but does not have finite complement. A correct specification of the cofinite topology can be found in Exercise 6 on page 43 in Section 3.1.
Page 69, Exercises
Exercises 2, 3, and 5 contain the ambiguous expressions \(\Fr A\cap U\), \(\Cl A\cap Y\), and \(\Fr A\cap Y\). The intended meanings are \( (\Fr A)\cap U\), \((\Cl A)\cap Y\), and \((\Fr A)\cap Y\).
In Exercise 4, the letter \(X\) is intended to denote \(\R^2\), the entire coordinate plane.
Page 78, Proof
The displayed formula has the wrong codomain. The formula should say \[g\circ f \colon X,\tau \to Z,\tau'' \] [correction contributed by Todd].
Page 78, Exercise 1
In the first part of the problem, about open intervals, the author allows intervals to be unbounded. But in the second part of the problem, closed intervals need to be bounded. In particular, the three sets \([0,1]\), \([0,\infty)\), and \(\R\) all are closed subsets of \(\R\), but no two of these sets are homeomorphic to each other.
Page 80, Definition 5
In the last line of the definition, the preposition “of” should be “to.”
Page 80, Proposition 17
The proposition is a special case of Exercise 3 on page 74.
Page 108, middle paragraph
Lacking is an explanation of why the existence of the open set \(V\) is significant. The point is that the closed set \(X\setminus V\) is both a subset of \(U(1)\) and a superset of \(U(0)\), so the closure of \(U(0)\) is necessarily a subset of \(U(1)\). Therefore Proposition 7 guarantees the existence of the required open set \(U(1/2)\).
Page 109, last paragraph
The rational numbers \(q\) and \(q'\) should be chosen to satisfy the additional property that the closed interval \([q,q']\) is a subset of the open interval \( (y_0 -p, y_0 +p)\). The reason for this requirement is that the set \(U(q')\) might contain a point \(x\) for which \(f(x)=q'\), so \(q'\) needs to be strictly less than \(y_0 +p\) to guarantee that \(f(V)\) is a subset of \( (y_0 -p, y_0 +p)\). Incidentally, the symbol \(V\) in use in this paragraph has a different meaning from the symbol \(V\) used in the middle of the preceding page.
Page 111, Example 16
The claimed difficulty in explicitly constructing \(F\) is no more than the difficulty in explicitly constructing \(f\), since \(F\) can be obtained from \(f\) simply by “connecting the dots.” If \(n\) is an integer, and the points \(f(n)\) and \(f(n+1)\) are known, then mapping the interval \( ( n, n+1)\) homeomorphically onto the line segment joining \(f(n)\) to \(f(n+1)\) is a routine matter (via a dilation, a translation, and a rotation).
Page 116, Section 6.2
The grammar in the third sentence of the first paragraph is doubtful. Since the set of positive integers is being considered as a single entity, not as a multitude, the appropriate verb is “does” instead of “do.”
Page 118, line 5
There is a typographical error: the closing curly brace after the word “is” should be a closing parenthesis.
Page 150, Example 7
In the sixth line, the punctuation mark after the word “however” should be a period (full stop), not a comma.
Page 157, Example 10
There is a typographical error in the fifth line: “other then” should be “other than.”
Page 158, Example 12
In the third line, the word “Then” should be “The.”
Page 160
At the end of the second full paragraph, “therefore assume that \(x\in S_i\) for some \(i\)” should say, “therefore \(S_i\in \mathcal{N}\) for some \(i\)”.
Page 162, Exercise 9
In the third line, the symbol p is in the wrong font.
Page 186
At the end of the second paragraph, the letter e is in the wrong font.
Page 186, Example 3
The displayed formula should say that \[ \Fr A = N \setminus A^\circ \supset N\setminus A \ne \varnothing. \]
Page 187, Exercise 2(d)
Insert a comma after the word “integer”.
Page 187, Exercise 6
In the second edition as originally published by Addison-Wesley, the exercise incorrectly asks for a proof that \(X\) is connected (which need not be the case). This error has been silently corrected in the Dover reprint.
Page 192, Exercise 6(d)
The point \( (0,1)\) presumably is intended to be the point \( (1,0)\).
Page 234, Figure 11.4
For \(f(x)\) read \(f(X)\), that is, the image under \(f\) of the whole space \(X\).
Page 235, Definition 1
There is a slight misuse of notation in the definition and throughout the section. The issue is that the domain of a function is an integral part of the function: two functions that have different domains are different functions. Accordingly, to say that the restriction of \(H\) to \(X\times\{1\}\) equals \(f\) makes no sense, for the domain of \(f\) is not \(X\times\{1\}\) but rather \(X\). Nonetheless, the intended meaning is clear: namely, \(H(x,1)=f(x)\) for every point \(x\) in the space \(X\). In other words, there is a natural bijection \(i\colon X \to X\times\{1\}\) that sends \(x\) to \((x,1)\), and \(H\circ i =f\).
Page 236, Example 3
An unstated assumption (implicit in Figure 11.6) is that the arcs \(a_1\) and \(a_2\) do not intersect except at the endpoints. Otherwise, something needs to be said about what “the area bounded by the images” means.
Page 238
The displayed formula five lines from the bottom of the page would be clearer with parentheses inserted: \[h\colon (X\times\{0\}) \cup (X\times\{1\}) \to Y \] (as in the displayed formula at the bottom of the page).
Page 241, second paragraph
In the second sentence, the phrase in quotation marks is missing a word and should say “is homotopic relative to \(y_0\) to.”
Page 242, proof of Proposition 2
Since \(f\sim g\) when \(f\) is the top edge of the homotopy and \(g\) is the bottom edge (by the author's definition), the roles of \(H'(r,0)\) and \(H'(r,1)\) are reversed: the former is \((a_3 \mysharp a_2)(r)\), and the latter is \((a_1 \mysharp a_2)(r)\). Figure 11.11 is correct.
Page 244, second paragraph
The indicated homotopy is actually between \( a_1 \mysharp {(a_2 \mysharp a_3)}\) and \( (a_1 \mysharp a_2)\mysharp a_3\), not the other way around (according to the definition of homotopy in Section 11.1). This discrepancy does not matter, since \(\sim\) has previously been shown to be a symmetric relation. Similarly, the labels on the vertices in Figure 11.12 are not consistent with the prior definitions.
Page 245, Exercise 3
In the second line, the left-hand parenthesis in \(\pi_1(Y,y_0)\) is defective, apparently a damaged piece of type from the pre-digital age.
Page 245, Figures 11.13 and 11.14
The lines across the middles of the figures have a small positive slope, but these lines instead should be horizontal.
Page 252, Figure 11.21
Reverse the directions of the arrows labeled \(j\) and \(j^{-1}\).
Page 256, Example 10
The two displayed formulas should say \[f\circ g = i_Y \] and \[g\circ f = k \sim i_X. \]