Are these spaces quotient spaces?

Question

I'm having trouble with this question:

Consider the following spaces:

1. Two circles of radius $1$ that are centered at $(0,1)$ and $(0,-1)$ (so that they touch at the origin - forming a "figure eight") with the subspace topology from $\mathbb{R}^2$. Let $f$ be any continuous map from $\mathbb{R}$ to this space. Can $f$ be a quotient map?

2. Consider the two sets $X$ and $Y$, equipped with the subspace topology from $\mathbb{R}^2$ below. Are they quotient spaces of the real line?

I'm not sure even where to start. What should I be looking for here?

Answer 1

Hint

Can $f:\infty\to\mathbb{R}$ be a quotient map? Well, $\infty$ is compact, $f$ is continuous. Can it be onto?

I denote by $\infty$ the union of the two circles.

Answer 2

Note that quotient maps are surjective so if you consider the continuous map $ f:R\rightarrow S^1\lor S^1$ given by $f(t)=e^{2\pi \iota t}+(1,0)$ is not surjective. Also constant map can also do this work.( Note also any continuous map $g:S^1\lor S^1\rightarrow R$ can not be surjective, since continuous image of compact set is compact, hence $g$ can not be quotient map)

Next $X$ is a quotient space of $R$ if you consider the partition of $R$ (any partition give raise to a equivalence relation, hence generates a quotient space) given by $$\{[1,\infty);(\infty,-1];\{x\};\{y\};\{z,-z\}\ | x\in (-1,-\frac{1}{2}); y\in (\frac{1}{2},1);z\in [-\frac{1}{2},\frac{1}{2}]\}$$

Similarly $Y $ is also a quotient space of $R$ if we consider the partition of $R$ given by $$\{[1,\infty);(\infty,-1];\{x\};\{y\};\{z\}\ \{\frac{1}{2},-\frac{1}{2}\}|x\in (-1,-\frac{1}{2}); y\in (\frac{1}{2},1); z\in (-\frac{1}{2},\frac{1}{2})\}$$

Note in both cases $[1,\infty)$,$(\infty,-1]$ are identified to two distinct points.We have used another idea to generate $Y$ ,namely if two endpoints of a closed-bounded interval of $R$ are identified then it is homeomorphic to $S^1$.