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.

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 $f^*(1+1)=f^*(0)=0$ […]

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 $\forall x,y \in \mathbb{R}, x\neq y \ \ \exists U,V \in \mathcal{T} \ : \ x\in U, y \in V, U \cap V = \emptyset$. […]

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 […]

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

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 […]

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 […]

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

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 […]

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 […]

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