32.803 general-topology questions.

1 Proving open set is equivalent across normed spaces

Let $A$ be an open set according to $\vert\vert \cdot \vert\vert^{A}$. Prove that $A$ is open according to a different norm $\vert\vert \cdot \vert\vert^{B}.$ Idea: Since $A$ is open, it follows ...

-2 Example of a set that is closed, bounded and it's boundary is not a set of measure $0$ [on hold]

I am looking for a subset $A$ of $\mathbb{R}$ that has the following properties: It's bounded It's closed It's boundary isn't of Lebesgue measure $0$ Edit: My problem was that i just couldn't think ...

1 Linear subspace of codimension one in infinite dimensional Banach space

Let $Y$ be a linear subspace of codimension 1 in an infinite dimensional Banach space $X$ i.e. $\dim (X/Y)=1$. Then how to prove that $X\setminus Y$ is path connected if and only if $Y$ is dense in $X$...

Prove $\{sum\ of\ line\ segments\ with\ ending\ points\ included\}\subset \Bbb{R}^2$ with euclidean metric is complete iff compact

$A\subset \Bbb{R}$ and let $X(A)\subset \Bbb{R}^2$ be a union of all segments connecting point $(0,1)$ with points $(a,0)$ such that $a\in A$. Show that $X(A)$ with euclidean metric is complete if and ...

1 What shape is homeomorphic to a sphere attached with a handle and a cross cap?

The classification theorem for closed surfaces states that a orientable closed surface is homeomorphic to a sphere with m handles with m >= 0; while a nonorintable closed surface is homeomorphic to a ...

3 “big” Hausdorff space with dense subspace of given cardinality

2 answers, 80 views general-topology set-theory cardinals
In a topology course we proved the following proposition: Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a dense ...

1 Continuous images of compact spaces

This question is related to An example of a compact topological space which is not the continuous image of a compact Hausdorff space?. Notation: A quasicompact space is one such that each open cover ...

2 {0,1} is not a retract of [0.1]

1 answers, 33 views general-topology homotopy-theory
Knowing that $Y\subset X$ is a retract of a topological space $X$ if there exists a continuous function $r:X\to Y$ such that $r(y)=y\quad\forall y\in Y$, I don't know how to show that $\{0,1\}$ is not ...

-1 Backward dense orbit implies forward dense is orbit.

0 answers, 30 views general-topology dynamical-systems
I have to determine if the next statement is true or false (in the context of differentiable dynamical systems) A homeomorphism with a backward dense orbit has a forward dense orbit I believe its ...