Articles of general topology

Is Max (R) a Hausdorff space?

Recall a space is totally disconnected if the only connected subsets are singletons (one-point subsets). Now let $R $ be a commutative ring with identity such that $\operatorname{Max}(R)$ is a totally disconnected space, in the sense of the Zariski topology. I want to know if $\operatorname{Max}(R)$ Hausdorff in this case?

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, let $\operatorname{diam} U$ denote the diameter, that is $$\operatorname{diam} U = \sup \{ \rho(x,y) : x,y \in U \} \,\,\,\,\, \operatorname{diam} \emptyset = 0 $$ […]

are singletons always closed?

I am learning about metric spaces and I find it very confusing. Is this a valid proof that a singleton must be closed? If $(X,d)$ is a metric space, to show that $\{a\}$ is closed, let’s show that $X \setminus \{a\}$ is open. Choose $y \in X \setminus \{a\}$ and set $\epsilon = d(a,y)$. Then […]

Topology – axioms of metric space – convergency – Cauchy

Let $(X,d)$ be a complete metric space and $U \subseteq X$, $U \neq X$, its open subset. Define a function $\rho\colon U \times U \rightarrow [0, \infty)$ as: $$\rho(x,y):=d(x,y)+\left|\frac{1}{d(x,X\setminus U)} – \frac{1}{d(y,X\setminus U)}\right|,$$ where $d(x,X\setminus U)$ is the usual distance between point $x$ and subset $X\setminus U$: i) Show that function $\rho$ satisfies the axioms […]

Clearing gap in Munkres

I’m working through 2-11 in Munkres on quotient topologies. They demonstrate that given a quotient map $q: X \to Y$ and continuous, quotient-respecting map $g: X \to Z$, there exists a continuous map $f: Y \to Z$ such that $f \circ p = g$ (this is theorem 11.1). I’m good here. But for a later […]

Orientability of Grassmannians

How can I understand when $Gr(n,k)$ is orientable and when not? I found that answer is yes if and only if $n \vdots 2$, but I do not know how to prove it.

Minimal Connected Set containing a Closed Connected Set in a Compact Space

This question came from Dugundji’s $\textit{Topology}$: Given a compact, connected space $X$, let $A \subset X$ be closed. Prove that there exists a closed, connected set $B \subset X$ such that $A \subset B$ and any proper subset of $B$ is either not connected, not closed, or does not contain $A$. The text has an […]

Why second countable for definition of manifold?

What is the motivation to define a manifold to be second countable? What kind of pedagogical issues does this avoid?

How to show that remainder $\beta X \setminus \beta(X)$ is an $F_\sigma$-set in $\beta X$.

The $\beta \mathbb N$ be the Stone-Čech compactification of $\mathbb N$. I have seen in Engelking – General topology that the remainder $\beta X \setminus \beta(X)$ is an $F_\sigma$-set in $\beta X$. Can anyone explain why? Thank you!

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What strategies could one follow in order to compute the number of path components of $Map(X,Y)$? I believe this number should also be the number […]