Are connected subspaces of connected subspaces are connected subsubspaces?

Question

Similar question in Algebra: Subsubgroups are subgroups of subgroups / Multiplicative Property of the Index

There, we had that for $A \subseteq B \subseteq C$ and $C$ is a group:

1. If $A$ is a subgroup of $B$ and $B$ is a subgroup of $C$, then $A$ is a subgroup of $C$.

2. If $A$ is a subgroup of $C$ and $B$ is a subgroup of $C$, then $A$ is a subgroup of $B$.

Now, I want to know for $A \subseteq B \subseteq C$ where $A, B$ and $C$ are topological spaces, do we have any of the following?

1. If $A$ is a connected subspace of $B$ and $B$ is a connected subspace of $C$, then $A$ is a connected subspace of $C$.

2. If $A$ is a connected subspace of $C$ and $B$ is a connected subspace of $C$, then $A$ is a connected subspace of $B$.

3. If $A$ is a connected subspace of $B$ and $A$ is a connected subspace of $C$, then $B$ is a connected subspace of $C$.

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Analogue for compactness: Are compact subspaces of compact subspaces are compact subsubspaces?

Answer 1

1 & 2 are true, and 3 is false.

1. Pf: Suppose on the contrary $\{D_1, D_2\}$ is a separation of $A$ in $C$, i.e. a partition of $A$ by clopen sets in $A$ as a subspace of $C$. By Exer 16.1 (*), $\{D_1, D_2\}$ are clopen sets in $A$ as a subspace of $B$. ↯ QED

2. Pf: Suppose on the contrary $\{D_1, D_2\}$ is a separation of $A$ in $B$, i.e. a partition of $A$ by clopen sets in $A$ as a subspace of $B$. By Exer 16.1 (*), $\{D_1, D_2\}$ are clopen sets in $A$ as a subspace of $C$. ↯ QED

3. $$[0,1] \subseteq [-1,5] \cup [6,10] \subseteq \mathbb R$$

Note: That $B$ is a connected subspace of $C$ is not used. Therefore, we have for $A \subseteq B \subseteq C$:

1. If $A$ is a connected subspace of $B$, then $A$ is a connected subspace of $C$.

2. If $A$ is a connected subspace of $C$, then $A$ is a connected subspace of $B$.

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(*)

Exer 16.1