Articles of general topology

Show the Euclidean metric and maximum metric are strongly equivalent.

I need to show that the Euclidean metric and maximum metric (or square metric??) are strongly equivalent. I have no idea how to start this proof. Any help? $d_1, d_2$ are called strongly equivalent if there exist positive constants $K, M$ such that for all $x, y\in X$: $Md_1(x,y)\leq d_2(x,y)\leq Kd_1(x,y)$

On continuously uniquely geodesic space II

This question was inspired by this answer of @wspin. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question : Is a complete uniquely geodesic space, continuously uniquely geodesic ?

Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set

I want to prove the following: Let $X$ be second countable zero-dimensional space. If $A,B \subseteq X$ are disjoint closed sets, there exist is a clopen set $C$ such that $A\subseteq C$ and $B\cap C = \emptyset $. (A topological space $X$ is zero-dimensional if it is Hausdorff and has basis consisting of clopen sets.)

A bijective continuous map is a homeomorphism iff it is open, or equivalently, iff it is closed.

Wikipedia states that “a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.”. How do we prove this fact? I can prove the obvious direction, but im unsure how to proceed the other ways

Is every $T_4$ topological space divisible?

A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the diagonal such that: $$V\circ V\subseteq U$$ Is every $T_4$ topological space divisible?

Proving Any connected subset of R is an Interval

Common Proof: Suppose $S$ is not an interval of $R$. Then by Interval Defined by Betweenness, $∃x,y∈S$ and $z\in R∖S$ such that $x<z<y$. Consider the sets $A_1=S∩(−∞,z)$ and $A_2=S∩(z,+∞)$. Then $A_1,A_2$ are open by definition of the subspace topology on S. Neither is empty because they contain x and y respectively. They are disjoint, and […]

Can the same subset be both open and closed?

This is a follow up response to: Counterexample to " a closed ball in M is a closed subset." I’m trying to understand this using only the given definitions of: Metric space, open/closed ball, open/closed subset. Let Metric $$M= \mathbb{Z} \cap [1,10]; d(x,y) = |x-y| $$ Let $P$ be the open ball: $P=B_{2.1}(5)=\{y \in M: […]

Prove that $\mathbb{N}$ with cofinite topology is not path-connected space.

$\mathbb{N}$ is the set of natural numbers. Let $U_{\alpha \in A} \subset \mathbb{N}$ be the subset such that its complement $\mathbb{N}$ \ $U_\alpha$ is a finite subset. Then $T= \{\emptyset, \mathbb{N}, U_\alpha |\alpha \in A \}$ is a topology on $\mathbb{N}$. Prove that $\mathbb{N}$ is not path-connected space with the following definitions below Definition: A […]

Evenly Spaced Integer Topology is Metrizable

Fustenborg’s proof uses an evenly spaced integer topology on $\mathbb Z$ which declares that a basis of open sets as those of the form $a + b \mathbb Z$ (i.e. arithmetic progressions). I’m interested in whether this space is metrizable. Pi-base claims that is, but I can’t understand what’s written. It says The topology on […]

Metric space and continuous function

Background: This is an exercise problem from Munkres’s Topology (Exercise 3 of Section 20 “The Metric Topology”, 2nd edition). It has been posted at this site: Topology induced by metric space. However, I am confused about some basic conceptual problems which have not been mentioned there. The Exercise: Let $X$ be a metric space with […]