Homework 02

• Topology (notebook version)

• Homework 2: Topology

  Due date:

  Student Name:

  Exercise 1.

  Find, for the case of the topology of the real line, an infinite intersection of open sets that is not open.

  Answer 1.

  Exercise 2.

  a.- Prove that a metric space has a topology induced by its metric in a manner similar to $\mathbb{R}$ in the previous example.

  b.- See that $d(x,y)=1$ if $x \neq y$ is a distance. What topology does this distance introduce?

  Answer 2. Exercise 3.

  1.- Let $(X,d)$ be a metric vector space, prove that:

  a.- $C^1_x = \{x'| d(x,x') \leq 1 \}$ is closed and is a neighborhood of $x$.

  b.- $N^{ϵ}_x =\{x' | d(x,x') < ϵ \}$, $ϵ >0$ is also a neighborhood of $x$.

  c.- $Int(N^{ϵ}_x) = N^{ϵ}_x$

  d.- $Cl(N^{ϵ}_x) = \{x' | d(x,x') \leq ϵ \}$

  e.- $\partial N^{ϵ}_x = \{x' | d(x,x') = ϵ \}$.

  2.- Let $(X,\cal{T})$ be a topological space and $A$ a subset of $X$. Prove that:

  a.- $A$ is open if and only if every $x \in A$ has a neighborhood contained in $A$.

  b.- $A$ is closed if and only if every $x$ in $A^c$ (i.e., not belonging to $A$) has a neighborhood that does not intersect $A$.

  3.- Let $(X,\cal{T})$ be a topological space, let $A \in X$ and $x \in X$. Prove that:

  a.- $x \in Int(A)$ if and only if $x$ has a neighborhood contained in $A$.

  b.- $x \in Cl(A)$ if and only if every neighborhood of $x$ intersects $A$.

  c.- $x \in \partial A$ if and only if every neighborhood of $x$ contains points in $A$ and points in $A^c$.

  Answer 3.

  Exercise 4.

  Review the proof of the following theorem:

  The map $ϕ:X → Y$ is continuous if and only if it holds that: given any point $x ∈ X$ and any neighborhood $M$ of $ϕ(x)$, there exists a neighborhood $N$ of $x$ such that $ϕ(N) ⊂ M$.

  Answer 4.

  Exercise 5.

  Let $ϕ : X → Y$ and $ψ : Y → Z$ be continuous maps, prove that $ψ ∘ ϕ : X → Z$ is also continuous. (Composition of maps preserves continuity.)

  Answer 5.

  Exercise 6. (optional)

  Prove that compact sets in the real line are closed and bounded.

  Answer 6.

  Exercise 7. (optional)

  a. Prove that if a space is Hausdorff then if a sequence has a limit point, it is unique.

  b. Let $X$ be compact, $Y$ be Hausdorff, and $ϕ: X → Y$ be continuous. Prove that the images of closed sets are closed.

  c. Find a counterexample if $Y$ is not Hausdorff.

  Answer 7.