Articles of general topology

Does the graph of a continuous function have an empty interior?

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous function and consider the set $\{ (x,f(x)) : x \in \mathbb{R} \}$ = G. Then, we obviously know that this set $G$ is closed. Now, I am having some difficulty trying to find its interior. My guess is that $\operatorname{Int} G = \varnothing $. But, how can I […]

The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite family $(a_j)_{j=1\dots m} \subset A$ and finite family $(r_j)_{j = 1\dots m} \subset \Bbb{R}_{\gt 0}$, define the “ball” […]

Brouwer's fixed point theorem (for unit balls) and retractions

Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if and only if $\partial B$ is not a retraction of $B$ proof. $(\Rightarrow)$ If $r: B\rightarrow \partial B$ is […]

Product of perfectly-$T_2$ spaces

Let $X$ be a topological space. Then, we say that $X$ is perfectly-$T_2$ or perfectly-hausdorff iff for every two distinct points $x_0,x_1\in X$, there is a continuous function $f\colon X\rightarrow [0,1]$ such that $\{ x_0\} =f^{-1}(0)$ and $\{ x_1\} =f^{-1}(1)$. That is, iff distinct points can be precisely separated by a continuous function. Is the […]

The set of which decimal expansion consists of 4 and 7

I’m sorry if my question is repeated. Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ contable? Is $E$ dense in $[0,1]$? Is $E$ compact? Is $E$ perfect? Proof: It is easy to see that $E$ is uncountable. Also $E$ is not dense […]

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of $\mathbb{R}^n$ for some $n$. Given a topological space $X$, we say that a subset $U$ of $X$ is numerically open if $\varphi^{-1}(U)$ is open […]

Symmetry of “is homotopic to” detail in the proof

Let $f,g:X\rightarrow Y$. If $f$ is homotopic to $g$ then $g$ is homotopic to $f$. Let $F:X\times I\rightarrow Y$ be a homotopy from $f$ to $g$ so $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x \in X$ and $F$ is continuous. Define $H:X\times I\rightarrow Y$ by $H(x,t)=F(x,1-t)$. We see $H(x,0)=g(x)$ and $H(x,1)=f(x)$. I am wondering what can […]

Homology and topological propeties

i have this theorem with it’s proof but i don’t understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where $\varphi^c=\lbrace x, \varphi(x)\leq c\rbrace$ Please thank you.

Why is $H_\delta^d$ monotone decreasing in $\delta$ in the definition of Hausdorff measure?

In trying to understand the definition of Hausdorff dimension I’m trying to get a good understanding of Hausdorff measure. On wikipedia they first make the following definition (found here): Let $(X,d)$ be a metric space, $S$ be any subset of $X$ and $\delta>0$ be a real number, then define $H_{\delta }^{d}(S)=\inf {\Bigl \{}\sum _{{i=1}}^{\infty }(\operatorname […]

Every quasi-compact scheme has a closed point

I know this question has been asked here before, but I have trouble understanding the following proof, taken from a Schwede’s write-up. I have underlined the bit I don’t understand. In particular, I have no idea how we can conclude that $P_3$ is not in $U_2$. Didn’t we pick $P_2$ to be any point in […]