Articles of general topology

Is the subset $ ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not compact as it can be expressed as union of the two disjoint open sets $[0,{\sqrt2}/{2}) $and$ ({\sqrt2}/{2}, \sqrt2)$ (though I’m not […]

Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”

As a consideration from the post “Prove by "elementary methods": The plane cannot be covered by finitely-many copies of the letter "Y"”, on the basis of the remark made in previous post by the user Moishe Cohen, is it still possible to apply elementary methods to prove weaker results, namely: The plane cannot be covered […]

Next book in learning General Topology

I have just finished the book “C Adams & R Franzosa – Introduction to Topology. Pure and Applied”. My aim is to reach to the level of the book “G E Bredon – Topology and Geometry”. Bredon’s book is not only too advanced to study after Adams’, but also I don’t think that it is […]

How can I prove that the Sorgenfrey line is a Lindelöf space?

How can I prove that the Sorgenfrey line is a Lindelöf space? Now, Sorgenfrey line is $\mathbb{R}$ with the basis of $\{[a,b) \mid a,b\in\mathbb{R}, a<b\}$, and in general, a topological space is called a “Lindelöf space” iff every open cover has a countable subcover. Please show me an elegant proof.

Possible areas for convex regions partitioning a plane and containing each a vertex of a square lattice.

If the plane is partitioned into convex regions each of area $A$ and each containing a single vertex of a unit square lattice, is $A\in (0,\frac{1}{2})$ possible? If each each vertex is in the interior of its region is $A \neq 1$ possible? More generally if $\rm{ I\!R}^n$ ($n\ge 1$) is partitioned into convex regions, […]

Topology and Arithmetic Progressions

I’m self-studying from “Elementary Topology Problem Textbook” by O.Ya.Viro et al. This is Exercise 2.Lx : Consider the following property of a subset $F$ of the set $\mathbb{N}$ of positive integers: there is $n$ ∈ $\mathbb{N}$ such that $F$ contains no arithmetic progressions of length $n$. Prove that subsets with this property together with the […]

determine the closures of the set k={1/n| n is a positive integer}

Consider the following topologies on $\mathbb{R}$: the standard topology the finite complement topology the lower limit topology. The question is to determine the closures of the set $k = $ {$\displaystyle \; \frac 1n|\; n \; \text{ is a positive integer }$} under each of these topologies. Please I would like somebody to help me. […]

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this condition? Thank you!

Distance between closed and compact sets.

This question is (1-21)(b) from M. Spivak’s Calculus on Manifolds. Question: If $A$ is closed, $B$ is compact, and $A \cap B = \emptyset$, prove that there is $d > 0$ such that $||y – x|| \geq d$ for all $y \in A$ and $x \in B$. Now, I interpret this as an instruction to […]

Question about Definition of homeomorphism (counter example)

I’m teaching my self topology with the aid of a book, but i’m confused about the meaning of homeomorphic. Below, I have 2 topologies, $\mathscr{T}_1$ and $\mathscr{T}_2$ and I’m pretty sure they are not homeomorphic, but I can make a continuous bijection between them. Definition of homeomorphism: If $X$ and $Y$ are topological spaces, a […]