Articles of general topology

Every separable metric space has cardinality less than or equal to the cardinality of the continuum.

Every separable metric space has cardinality less than or equal to the cardinality of the continuum. This was stated in my book in a series of questions using the post office metric on $\Bbb R^2$, but I can’t think of a way to prove this.

Continuous map from Projective Plane to Torus

Is there a continuous map from the $\mathbb{R}P^2$ to the torus which is not homotopic to a constant map? My working: I am pretty sure the answer is no. But I am not sure how to present the answer “properly”. Is it correct to say that any map $f:\mathbb{R}P^2\to T$ induces $f^*:\mathbb{Z}/2\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}$. $f^*(1+1)=f^*(1)+f^*(1)$ But $f^*(1+1)=f^*(0)=0$ […]

Lower limit topology is a Hausdorff space $T_2$

Could you help me with the following? Consider lower limit topology $\mathcal{T} $ with basis $\mathcal{B} = \{ [a,b) \ | \ a,b \in \mathbb{R}, \ a<b \}$ Show that $\forall x,y \in \mathbb{R}, x\neq y \ \ \exists U,V \in \mathcal{T} \ : \ x\in U, y \in V, U \cap V = \emptyset$. […]

Proving that a quotient space is compact but not Hausdorff

Let ∼ be the equivalence relation on $\mathbb{R^2}$ defined by $(x, y) ∼ (x_0 , y_0 )$ if and only if there is a nonzero $t$ with $(x, y) = (tx_0 , ty_0$ ). Prove that the quotient space $\mathbb{R^2}/ ∼ $ is compact but not Hausdorff. My attempt : $\mathbb{R^2}/ ∼ $ is homeomorphic […]

Prove that these loops are homotopic

This question already has an answer here: Topological group: Multiplying two loops is homotopic to linking these paths? 2 answers

Proof category of $k$-spaces is “almost” locally Cartesian closed

From this comment on this MO question about locally Cartesian closed subcategories of topological spaces I understand the following proposition holds. Proposition. Let $\mathsf C$ be the category of $k$-spaces. If $A,B$ are compactly generated, then $f^\ast :\mathsf C_{/B}\to \mathsf C_{/A}$ admits a right adjoint. The comment directs to May-Sigurdsson, but it’s terse for me […]

When is a compact of the plane included in a connected compact with empty interior?

The question is pretty much in the title : are there any nice conditions for a compact $K$ of the plane to be included in some other connected compact with empty interior ? One obvious necessary condition is to have empty interior, so we can assume that it is the case. One obvious sufficient condition […]

Real Projective Space, $\mathbb P^n$, is Hausdorff.

This question already has an answer here: How can I prove formally that the projective plane is a Hausdorff space? 7 answers

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A “compact” set is one where every open cover has finite subcover. Is there such a thing as a set that is covered by an infinite cover of open subsets, and what would it be […]

Baby Rudin Problem 2.16

Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q) = |p-q|$. Let $E$ be the set of all $p\in Q $ such that $2<p^2<3$. Show that $E$ is closed and bounded in $Q$, but $E$ is not compact. Is $E$ open in $Q$? To show $E$ is closed I thought […]