Erratum: Geometric Aspects of General Topology

The online version of the original book can be found at http://dx.doi.org/10.1007/978-4-431-54397-8

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The original version of the book had typos and incorrect symbols/characters which have been fixed in the respective chapters of this book.

Preface

p. vii, line 9 from top, 1966 should read as 1967

Chapter 1

p. 1, line 12 from bottom: Insert “ — half line” before ‘;’.

p. 2, line 7 from top: cellurality should read as cellularity

p. 19, line 9 from top: $n \in \varGamma$ should read as $n \in \mathbb{N}$

Chapter 2

p. 47, Fig. 2.7 should read as Fig. 2.8. This figure should be on p. 48

p. 47, line 5 from bottom: Fig. 2.8 should read as Fig. 2.7

p. 48: Fig. 2.8 should read as Fig. 2.9. This figure should be on p. 50

p. 48, line 13 from bottom: Remove “— Fig. 2.9”

p. 48, line 1 from bottom: Insert “— Fig. 2.8” before ‘.’

p. 49, line 10 from bottom: (Fig. 2.7) should read as (Fig. 2.9)

p. 50: Fig. 2.9 should read as Fig. 2.7. This figure should be on p. 47

p. 51, line 1 from top: Remove “Let X be a paracompact space.”

p. 51, line 3 from top: Insert “Let A be a subspace of X.” before ‘To find ...’

p. 65, line 7 from bottom: with (1) should read as with (2)

Chapter 4

p. 137, line 7 from bottom: call should read as called

p. 156, line 11 from top: fs should read as f is

p. 160, line 10 from bottom: polyhedra should read as polyhedron

p. 186, line 4 from top: ${K}^{{\prime}}\!\left (0\right )$ should read as ${K'}^{\left (0\right )}$

p. 187, line 2 from top: Insert “If $x \in {K}^{\left (0\right )}$ then K _x  = K.”

Chapter 5

p. 249, line 3 from top: Insert “dim X” after ‘dimension’

p. 249, line 4 from top: n + 1. and should read as n + 1, and

p. 254, line 15–21: This proof is only for the case X and Y are closed in ${\mathbb{R}}^{n}$.

For the general case, the proof should be written as follows:

Proof. For each homeomorphism h : X → Y , we will show that $h\!\left (\mbox{ int}\ X\right ) \subset \mbox{ int}\ Y$. Then, applying this to the inverse homeomorphism h ⁻¹ : Y → X, we can also obtain ${h}^{-1}\left (\mbox{ int}\ Y \right ) \subset \mbox{ int}\ X$, that is, $\mbox{ int}\ Y \subset h\!\left (\mbox{ int}\ X\right )$. Thus, we will have $h\!\left (\mbox{ int}\ X\right ) = \mbox{ int}\ Y$.

To see $h\left (\mbox{ int}\ X\right ) \subset \mbox{ int}\ Y$, note that each x ∈ int X has a compact neighborhood C in ${\mathbb{R}}^{n}$ with C ⊂ X. Since $\mbox{ int}\ h\left (C\right ) \subset \mbox{ int}\ Y$, we may show that $h\left (x\right ) \in \mbox{ int}\ h\left (C\right )$. On the contrary, assume that $h\left (x\right ) \in \mbox{ bd}\ h\left (C\right )$. For each neighborhood U of x in C, $h\left (U\right )$ is a neighbourhood of $h\left (x\right )$ in $h\left (C\right )$. We can apply Theorem 5.1.7 to find a neighbourhood V of $h\left (x\right )$ in $h\left (C\right )$ such that $V \subset h\left (U\right )$ and every map $g : h\left (C\right )\setminus V \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$ extends to a map$\mathop{\tilde{g}} : h\left (C\right )\setminus V \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$. Then, ${h}^{-1}\left (V \right )$ is a neighborhood of x in C with ${h}^{-1}\left (V \right ) \subset U$. For every map $f : C\setminus {h}^{-1}\left (V \right ) \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$, $f{h}^{-1} : h\left (C\right ) \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$ can be extended to a map $\tilde{f} : \mbox{ }h\left (C\right ) \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$. Then, $\tilde{f}h : C \rightarrow {\mathrm{\mathbf{S}}}^{n-1}$ is an extension of f. Due to Theorem 5.1.7, this means that x ∈ bd C, which is a contradiction. Therefore, $h\left (x\right ) \in \mbox{ int}\ h\left (C\right )$.

p. 261, line 6 from bottom: f ⁻¹ should read as h ₀ ⁻¹

p. 263, line 14 from top: Insert the following at the end of the sentence:

Corollary 5.2.16 is valid even if n = ∞. In fact, $\left (\mbox{ pr}_{i}^{-1}\left (0\right ),\mbox{ pr}_{i}^{-1}\left (1\right )\right )_{i\in \mathbb{N}}$ is essential in ${\mathrm{\mathbf{I}}}^{\mathbb{N}}$. This will be shown in the proof of Theorem 5.6.1.

p. 264, line 6 from top: Insert “and” between ‘CHARACTERIZATION’ and ‘the’.

p. 264, line 7 from top: Insert “respectively” after ‘dimension’

p. 268, line 12 from top: Since should read as Note that $\mathcal{U}_{i}$

p. 268, line 12 from top: it should read as $\mathcal{U}_{i}$. Then, it

p. 293, line 16 from bottom: Y should read as ${\mathbb{R}}^{2n+1}$

p. 316, line 6 from bottom: $\upvarepsilon /2$ should read as $\upvarepsilon /3$

p. 319, line 13 from top: $n \in \mathbb{N},$ and should read as and $n \in \mathbb{N}$. For any infinite set

p. 319, line 14 from top: Delete ‘such that … infinite. Then’.

p. 320, line 6 from bottom: B ₁ should read as B ₁ in ${\mathrm{\mathbf{I}}}^{\mathbb{N}}$.

p. 320, line 6 from bottom: Replace ‘which implies that’ by the following:

By Lemma 5.3.7, if P is a partition between A ₁ ∩ S and B ₁ ∩ S in S, then there is a partition P ^′ between A ₁ and B ₁ in ${\mathrm{\mathbf{I}}}^{\mathbb{N}}$ such that P ^′∩ S ⊂ P. Then, it follows that P ≠ ∅. Due to Theorem 5.2.17, this means that $\dim S \geq 1$, that is,

Chapter 6

p. 346, line 11 from bottom: homotopy should read as deformation

p. 346, line 10 from bottom: Delete ‘h ₀ = id and’.

p. 346, line 1 from bottom: Add the following:

It is said that X is deformable into $A\left (\subset X\right )$ if there is a deformation h : X ×I → X with $h_{1}\left (X\right ) \subset A$. A retract A of X is a deformation retract of X if X is deformable into A (refer 6.2.10(9)).

p. 348: Insert the following before Section 6.3:

(9) A subset A of a space X is a deformation retract if and only if X is deformable into A and A is a retract of X.

To see the “if” part, let h : X ×I → X be a deformation with $h_{1}\left (X\right ) \subset A$ and let r : X → A be a retraction. Using the fact that r h ₁ = h ₁, we can define a homotopy from id _X to r.

p. 363, line 5 from top: Add “as a closed set” after ‘Banach space)’.

p. 371, line 5 from top: 4.9.10 should read as 4.9.11

Index

p. 516, right-side line 2 from bottom: cellurality should read as cellularity

p. 518, left-side line 12 from top: hedgehog, 33 should read as hedgehog, 33, 296