Articles of general topology

A direct proof that a compact metric space is sequentially compact

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks

The range of a continuous function on the order topology is convex

Let $(W, \leq)$ be a linear order, and let $f : [0, 1] \rightarrow W$ be a continuous function (where [0, 1] has the usual topology and W has its order topology). Show that the range of f is convex. Let $Y \subseteq X$. The subspace $Y$ is a convex set if for each pair […]

If a set contains all its limit points must it be closed?

If a set $X$ in a topological space $T$ has the property that for all sequences $x_n \in X, x_n \to x \implies x\in X$ must X be closed? I know this is true for metric spaces but is it true for a general topological space. Here I am defining $X$ is closed iff $T\setminus […]

$H_2$ and $\pi_1$ of open subsets of $\mathbb{R}^2$

If $U$ is an open subset of $\mathbb{R}^2$, is it true that $H_2(U)=0$ and what can we say about $\pi_1(U)$? (For example, can we show that $\pi_1$ isn’t perfect, e.g. $\pi_1(U)\neq 0\Rightarrow H_1(U)\neq 0$?) For $\pi_1$, I’m aware that there is an answer here: fundamental groups of open subsets of the plane, but I’d rather […]

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing we have to worry about is optimization. This is why my question probably isn’t as […]

Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice

There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?

What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

This originally comes from $f_1(x,y)=\frac{x}{y}$, where $X=\mathbb{R}^{n}, Y=\mathbb{R}^m, x \in X, f: X \rightarrow Y, x \neq 0, f(x) \neq 0$ $$\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$$ If I understand it correctly, $x$ and $y$ can be anything but zero, and $h_x,h_y$ go towards zero. Moreover, both numerator and denominator cannot be negative. But since $x,y$ could be […]

Show that the lexicographic order topology for $\mathbb{N}\times \mathbb{N}$ is not the discrete

My question is: Show that the lexicographic order topology for $\mathbb{N}\times \mathbb{N}$ is not the discrete I have been thinking on the fact that on the discrete topology all singleton sets are open. If can I find one singleton not open the proof is done? For example $\{(0,0)\}$ is not open because it has not […]

Typo or additional hypotheses needed in problem 5, p.71 of Bredon's Topology and Geometry?

The problen can be foun in p.71 of Topology and Geometry. I state it below for convenience. Problem: Consider the half open real line $X=[0,\infty)$. Define a functional structure $F_{1}$ by taking $f\in F_{1}(U) \Leftrightarrow f(x)=g(x^{2})$ for some $C^{\infty}$ function $g$ on $\{x| x\in U\text{ or} -\negmedspace x\in U\}$. Define another functional structure $F_{2}$ by […]

Density character of a subspace of a topological space.

Let $(X,\tau)$ be a topological space. Suppose $dc(X)=\kappa$ and let $D\subset_{dense} X$ be a dense subset of $X$ of cardinality $\kappa$. Is it true that $X\setminus D$ has density character $\kappa$, as a subspace of $X$ with the restricted topology?