Articles of general topology

Difference between complete and closed set

What is the difference between a complete metric space and a closed set? Can a set be closed but not complete?

Unnecessary property in definition of topological space

A set $X$ with a subset $\tau\subset \mathcal{P}(X)$ is called a topological space if: $X\in\tau$ and $\emptyset\in \tau$. Let $L$ be any set. If $\{A_\lambda\}_{\lambda\in L}=\mathcal{A}\subset\tau$ then $\bigcup_{\lambda\in L} A_\lambda\in\tau$. Let $M$ be finite set. If $\{A_\lambda\}_{\lambda\in M}=\mathcal{A}\subset\tau$ then $\bigcap_{\lambda\in M} A_\lambda\in\tau$. Let $\emptyset=\mathcal{A}=\{A_\lambda\}_{\lambda\in N}$, i.e $N=\emptyset$. Then by 2: $$\bigcap_{\lambda\in N} A_\lambda=\{x\in X; \forall […]

On convergence of nets in a topological space

Let’s consider a topological space that is not necessarily metrizable. Question: I wonder what is the motivation for defining convergence of nets in a topological space? What do we gain in working with convergence of nets rather than convergence of sequences?

Is the union of finitely many open sets in an omega-cover contained within some member of the cover?

Let $\mathcal{U}$ be an open cover of $\mathbb{R}$ (Standard Topology) such that $\mathbb{R} \not \in \mathcal{U}$ and for any finite set $A$ there is a $U \in \mathcal{U}$ such that $A \subseteq U$. We call such an open cover an $\omega$-cover. Can we show that for any finite set $B \subset \mathcal{U}$, there is a […]

Uncountable product of separable spaces is separable?

This question already has an answer here: On the product of $\mathfrak c$-many separable spaces 2 answers

If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant

If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant. Thoughts: I can see that this is similar to the definition of connectedness: that any continuous map from $X$ to a two-point space is constant. How would I go about proving […]

Why are these two definitions of a perfectly normal space equivalent?

I’ve been skimming through some topology textbooks recently. Some sources, (such as Munkres’ Topology and Willard’s General Topology) define a space $(X,\mathcal{T})$ to be perfectly normal iff $X$ is normal and every closed set is a $G_\delta$ set, that is, a countable intersection of open sets. Other sources (such as Dudley’s Real Analysis) define a […]

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, “without $\Bbb R$”* so that it Reduces to the traditional definition when desired? Has the same use in at least some of the higher contexts where we would use the present differentiable manifolds? Motivation/Context: I was a little bit dissapointed when I learned […]

The set of lines in $\mathbb{R}^2$ is a Möbius band?

I encountered a hard-to-believe and hard-to-understand statement in this problem: Let $\mathcal{L} = \{\textrm{lines in}\ \mathbb{R}^2\}$. Consider the 2-to-1 map $f: S^1 \times \mathbb{R} \rightarrow \mathcal{L}$ given by $(\theta,x) \mapsto L_{\theta,x} = \{t(\mathrm{cos}\,\theta,\mathrm{sin}\,\theta)+x(-\mathrm{sin}\,\theta,\mathrm{cos}\,\theta): t\in\mathbb{R}\}$. Show that for almost every $(\theta,x) \in S^1\times\mathbb{R}$, we have that $f^{-1}(L_{\theta,x})$ is a smooth submanifold of $X$. Remark: The set […]

Locally Constant Functions on Connected Spaces are Constant

I am trying to show that a function that is locally constant on a connected space is, in fact, constant. I have looked at this related question but my approach is a little different than the suggested approach and I’m unsure about the final step and would appreciate a tip. Here is what I have […]