Articles of general topology

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset is compact. I know that all finite dimensional inner-product spaces are equivalent, but I never heard it about metric so my hunch […]

homeomorphism from $\mathbb{R}^2 $ to open unit disk.

I want to show that $\mathbb{R}^2$ and the open unit disk are homeomophic. This seems obvious as I can easily stretch the disk into the real plane, but I cannot think of a way to analytically do it. Does $$f(x,y)=(f_1(x,y),f_2(x,y)) =(\frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}). $$ get the job done?

Rudin's PMA Exercise 2.18 – Perfect Sets

This question already has an answer here: Perfect set without rationals 7 answers

Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don’t know why $X$, the set of all outcomes generated by a fixed strategy of one player, constitutes a perfect set. I can see it’s true, if the fixed strategy is a constant […]

Compact connected spaces have non- cut points

Let $X$ be a compact connected Hausdorff space with more than one point. Prove that there is point $x \in X$ s.t. $X \setminus \{x\}$ is connected.

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ such that $p(y_0)=f(x_0)$, there is a unique continuous function (a “lift”) $\tilde f:X\to Y$ such that $p\circ\tilde f=f$ and $\tilde f(x_0)=y_0$. […]

Why is $^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point definition? Also I tried to show that $\beta\mathbb{N}$ has no non constant converging sequences but ran into some trouble. How do you show […]

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, we have that $(f(x_n))$ accumulates only on $\partial D_2\cup\{\infty\}$. [$\partial D$ denotes boundary of $D$.] Definition: A map $f:X \to Y$ is called proper if $f^{-1}(K)$ […]

What does continuity of inclusion means?

If $A,B,C$ are three spaces such that $A\subset B\subset C$ and $A$ is dense in $C$. Now my teacher said that the inclusion between three spaces are continuous and so you can directly say that $B$ is dense in $C$ . When i asked him why can’t you directly conclude this, he said that the […]

Is uncountable subset of separable space separable?

I have to prove that any uncountable $B\subseteq \mathbb{R}$, where $(\mathbb{R},\epsilon^1)$ is euclidean topology and topology on B is relative, is separable. And I know it’s true because every subset of separable metric space is separable. But what if we are given separable space $(X,\tau)$, $X$ uncountable, and $A \subseteq X$ uncountable subset with relative […]