Topology

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Problem: Let's prove that the set of all intervals of \( R \) with rational ends is countable.

16.1 Topology

5.14 $

Problem: Can the intersection of two perfect sets be imperfect?

16.2 Topology

2.57 $

Problem: Show that any metric space satisfies the first axiom of countability in any of its points.

16.3 Topology

2.57 $

Problem: Show that any group with a discrete topology is a topological group.

16.4 Topology

2.57 $

Problem: Let's show that for any sets \( X, Y, Z \) the equalities are true: a) \( X \times(Y \cup Z)=(X \times Y) \cup(X \times Z) \), b) \( (X \cap Y) \times Z=(X \times Y) \cap(Y \times Z) \)

16.5 Topology

1.03 $

Problem: Show that if \( \rho(x, y) \) is the metric on the space \( X \), then \( \tan ^{-1} \rho(x, y) \) is also a metric on \( X \).

16.6 Topology

2.06 $

Problem: Let \( X \) is an infinite set, and \( \mathcal{T} \) is the set of those subsets of \( X \), the complement of which is finite. Show that \( \mathcal{T} \) is a topology.

16.7 Topology

5.14 $

Problem: Show some subgroups \( \mathbb{R} \) and \( \mathbb{R}^{n} \).

16.8 Topology

1.28 $

Problem: Let's prove the relation \( X \times Y=A \times B \Leftrightarrow X=A \), \( Y=B \).

16.9 Topology

0 $

Problem: Show that if \( \rho(x, y) \) is a metric on the space \( X \), then \( \frac{\rho(x, y)}{1+\rho(x, y)} \) is also a metric on \( X \).

16.10 Topology

2.06 $

Problem: Show that there exists a countable subspace \( \mathbb{R}^{2} \), which is a discrete space.

16.11 Topology

2.57 $

Problem: Show that for any \( a \in \mathbb{Z} \) the set \( \mathbb{Z}_{a}=\{z \in \mathbb{Z}: z \) is divisible by \( a\} \) is a subgroup of \( \mathbb{Z} \).

16.12 Topology

1.28 $

Problem: Let the set \( U \) is open and everywhere dense on the metric space \( X \). Show that \( X / U \) is nowhere dense on \( X \).

16.13 Topology

2.57 $

Problem: Show that if a space has a countable base, then every base of this space contains a countable family, that is a base.

16.14 Topology

4.37 $

Problem: Is the group \( \mathbb{Z} \) a subset of the topological group \( \mathbb{R} \) ? \( \mathbb{Q} \) ?

16.15 Topology

3.08 $