Articles of quotient spaces

Quotient space of unit sphere is Hausdorff

Let $S^1=\{e^{2\pi it}|t\in\mathbb{R}\}$ be the unit sphere. Define $\sim$ on $S^1$ where two points are identified if $t_1-t_2=k$, for some $k\in\mathbb{Z}$. It must be shown that $S^1/\sim$ is Hausdorff. The quotient space of a topological space is Hausdorff if its graph is closed. That is, if $R=\{(x,y)\in S^1\times S^1|x \sim y \}$ is closed, then […]

Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.

As the title says, I’m trying to prove that if $X/R$ is a Hausdorff space then $R\subset X\times X$ is closed. I have several questions about this: $(1)$ What exactly is $R$? I thought of $R$ as an equivalence relation, but I never thought of that relation (or any) as a set. What are the […]

difference between a $G$ invariant measure on $G/H$ and a haar measure on $G/H$

Let $G$ be a locally compact topological group, and $H$ be a normal subgroup. $G/H$ is a locally-compact topological group as well, and if we assume $H$ to be closed then $G/H$ is hausdorff and therefore admits a haar measure $\mu_{G/H}$. It is also known that if $\Delta_{G}|_{H}=\Delta_{H]$ (the modular functions), then $G/H$ admits a […]

Let $W$ be a subspace of a vector space $V$ . Show that the following are equivalent.

Show that $\textbf{v} + W \subseteq W \Rightarrow W \subseteq \textbf{v} + W.$ Here is my proof, is it correct? Is there any easier way? Note that for any element $\mathbf{v + w}$ in $v+W$, $\mathbf{v+w}$ is in $W$ as well. Hence $\mathbf{v} + \mathbf{w} = \mathbf{w}_{1}$ for some $\mathbf{w_1} \in W$ Which implies $\mathbf{v} […]

A Map From $S^n\to D^n/\sim$ is Continuous.

The following question is motivated by Thomas’s answer here which can be used to prove that $\mathbf RP^n$ is same as the space obtained by identifying the antipodal points on the boundary circle of the closed $n$-disc. Notation: Let $H^n_+$ denotes the open upper half-sphere of $S^n$, $\bar H^n_+$ denote the closed upper half-sphere, and […]

Understanding quotient groups

Admittedly this will be probably be a naive question, but here it goes: Is it possible to flesh out in simple terms, for someone with little background in group theory, what it means to take the quotient group of $\mathbb{R}$ or $\mathbb{C}$ by a lattice $\Lambda$ or by $\mathbb Z$? How to evaluate this quotient? […]

Quotient topology by identifying the boundary of a circle as one point

The following is an example taken from Munkres topology book: I don’t understand why does $X^{*}$is homeomorphic to $S^{2}$, is this a basic fact that I don’t understand or is it an example of something more advanced ? Also, I don’t understand figure 22.4, I think that a saturated set is either contained in $\{(x,y)\mid […]

Quotient space of the reals by the rationals

Let $\mathbb{R}/{\sim}$ be the quotient space given by the equivalence relation $a \sim b$ if $a$ and $b$ are rational. I am trying to understand general properties of the quotient topology and this example seems worth fleshing out in full. It’s also a very strange example to me so I’d appreciate feedback on what I’ve […]

A relation between product and quotient topology.

I was studying a topic about Algebraic Topology and a question popped into my mind: Suppose that we have two topological spaces $X$ and $Y$. Let $\sim_X$ and $\sim_Y$ equivalence relations in X and Y. In $X\times Y$, we can define the following equivalence relation: $$(x,y) \sim (x’,y’)\ \ \mbox{when}\ \ x\ \sim_X\ x’ \ […]

The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$

Define $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$. By identifying all the points of $S^{n-1}$ we get a topological space which is intuitively homeomorphic to $S^n$. If $n = 2$, this can be visualised by pushing the centre of the disc $D^2$ down so you have a sack, then shrinking the […]