Articles of general topology

Gaps in my proof of the Arzela-Ascoli Theorem – help and expertise greatly appreciated for an alternate formulation.

I have a general outline of the proof of the Arzela-Ascoli Theorem but have trouble filling in the gaps of the theorem. I have posted the entire general method I believe to be correct below. I was wondering if there was an easier way to do this or if someone would know of an easier […]

Prove that the countable complement topology is not meta compact?

I have seen the proof of (countable complement topology is not meta compact) , which says that the countable intersection of open sets is open and thus uncountable, so this topology cannot be meta compact. It is easy to see that the countable intersection is open and uncountable, but why this implies that this topology […]

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open sets such that for open $U,V \in X$: $\ Hom_{Op_X}(U,V) = \begin{cases} \{pt\} & U \subset […]

Show that the set of isolated points of $S$ is countable

Let $S$ be a subset of $\mathbb{R}^n$; show that the set $I$ of isolated points of $S$ is countable. Let $\mathbf{x}\in I$. There exists an open ball, say $B(\mathbf{x},r_\mathbf{x})$, of radius $r_\mathbf{x}$, for each $\mathbf{x}$ such that $B(\mathbf{x},r_\mathbf{x}) \cap S=${$\mathbf{x}$}. Now without change of notation replace $ r_\mathbf{x}$ with $\frac{1}{2}r_\mathbf{x}$. Thus the open balls $B$ […]

Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff

This question already has an answer here: $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed 4 answers

Showing that metric induces single unique topology on a finite set

I am trying to prove, that given a metric on a finite set it induces exactly one topology. I have an idea which might lead to a proof, but am not sure: For a finite set X with a given metric d we can prove it is a discrete topology: $\forall x \in X$ take […]

General Topology and Basis definition

In the book “Introduction to Smooth Manifolds” by John M. Lee, the author defined general topology and basis as: A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$, called open sets, satisfying: $X$ and $\emptyset$ are open The union of any family of open sets is open The intersection of […]

liminf in terms of the point-to-set distance

Let $\mathcal{X}$ be a normed space and $C\subseteq \mathcal{X}$. We define the point-to-set distance for the set $C$ to be: $$ d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in [0,\infty] $$ Additionally, we define the inner limit of a sequence of sets $C_n$ in $\mathcal{X}$ to be: $$ \liminf_n C_n = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty C_m $$ This […]

Topology of the ring of formal power series

I’m interested in defining a topology on the ring $R[[X_i]]$ of formal power series in $(X_i)_{i\in I}$, where $R$ is a topological ring and $I$ is a (possibly infinite) index set. The wiki article discusses several options for this, and it seems that the most natural topology satisfies $(x_n)_{n\in\infty}$ converges iff for every monomial $X^\alpha$ […]

Locally Euclidean Hausdorff topological space is topological manifold iff $\sigma$-compact.

I’d like somebody to specify flaws in my outline of the proof of the above statement. I’m following the definition of topological manifold used in Lee’s Introduction to Smooth Manifolds. (it is 2nd countable) Let $X$ to be a Locally Euclidean Hausdorff topological space. $\Rightarrow$: If $X$ is a topological manifold, then it has a […]