Math Problems Vault

Problem 15.30 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 15.4 "Continuous Mappings, Weak Topologies, and Metrizability"

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ takes a minimum value. Show that any topological space that is homeomorphic to $X$ also possesses this property.

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Problem 15.31 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 15.4 "Continuous Mappings, Weak Topologies, and Metrizability"

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ has an interval as its image. Show that any topological space that is homeomorphic to $X$ also possesses this property.

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Problem 15.32 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 15.4 "Continuous Mappings, Weak Topologies, and Metrizability"

Show that $\mathbb {R}$ is homeomorphic to the open bounded interval (0,1), but is not homeomorphic to the closed bounded interval $[0,1]$ .

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Problem 15.50 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 15.6 "Connected Topological Spaces"

(i)(ii)

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(i)

Show that a closed, bounded interval is not homeomorphic to an open bounded interval.

(ii)

Show that $[0,1)$ is not homeomorphic to $(0,1)$ .

Problem 1.6.9 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.6 "Connectedness"

Show that the letter T is not homeomorphic to the letter O.

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Remark.

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Consider what happens when a point is removed from each letter and the corresponding connectivity properties.

Problem 1.6.10 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.6 "Connectedness"

Show that $S^1$ is not homeomorphic to $\mathbb {R}$ by showing $S^1 \setminus \left\{ x\right\}$ is not homeomorphic to $\mathbb {R} \setminus \left\{ y\right\}$ .

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Problem 1.6.11 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.6 "Connectedness"

Show that two disjoint concentric circles in the plane are not homeomorphic to one circle.

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Problem 1.9.66 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

(a)(b)

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(a)

Show that the following two subsets of $\mathbb {R}$ are not homeomorphic: $A = \left\{ 1/n : n \in \mathbb {N}\right\} \cup \left\{ 0\right\}$ , $B = \mathbb {N}$ .

(b)

Show that $B^+$ is homeomorphic to $A$ , where $B^+$ denotes the one-point compactification of $B$ .

Problem 1.9.67 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Show that $\infty$ (the figure-eight curve) is not homeomorphic to O (the circle).

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Remark.

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Consider where the crossing point of $\infty$ could go under a homeomorphism.

Problem 1.9.68 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

The next four exercises concern the homeomorphism type of the letters of the alphabet. In each case, assume the letter is written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ .

Show that the letter X is not homeomorphic to the letter Y, but that the letter Y is homeomorphic to the letter T.

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Problem 1.9.70 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

The letters of the alphabet are written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ .

Prove that the letter A is not homeomorphic to the letter B.

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Problem 1.9.71 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

The letters of the alphabet are written in the sans serif style, with no adornments: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ .

Group the letters of the alphabet into equivalence classes so that equivalent letters are homeomorphic and nonequivalent letters are not homeomorphic.

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Problem 1.9.74 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Show that the torus is not homeomorphic to an open set in $\mathbb {R}^{2}$ .

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Remark.

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Use the properties of compactness and connectedness.

Problem 1.9.75 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Show that if $A \subset S^1$ with $A \neq S^1$ , then $A$ is not homeomorphic to $S^1$ .

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Problem 1.9.87 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Show that there is no homeomorphism of the plane to itself which sends the unit circle to itself and sends $(0, 0)$ to $(2, 0)$ .

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Problem 1.9.91 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Consider the two pairs of circles $A \cup B$ and $C \cup D$ from the figure in Exercise 1.9.90, with $x \in A$ and $y \in C$ the marked points.

Show that any homeomorphism sending $A \cup B$ to $C \cup D$ which sends $x \in A$ to $y \in C$ must send $A$ homeomorphically to $C$ and $B$ homeomorphically to $D$ .

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Problem 1.9.92 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 1.9 "Supplementary exercises"

Consider the two pairs of circles $A \cup B$ and $C \cup D$ from the figure in Exercise 1.9.90.

Show that there does not exist a homeomorphism of the plane sending $A \cup B$ to $C \cup D$ .

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Remark.

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Consider the regions bounded by $A$ and $C$ .

Homeomorphism invariants and non-homeomorphic spaces

References