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How do you know whether a set is connected?

Please give me a sufficient answer to this question, I am not very familiar with this and also I am unable to find a comprehensive book in this domain.

A:

It doesn’t matter.
Suppose you have a space $X$ with a topology $\tau$, and a subset $A$ of $X$.
It is a basic property of a topological space that the union of all open sets which contain a point of $A$ is also in $\tau$, i.e., $A\subseteq \bigcup \{ O \ | \ O \in \tau, \ A \cap O eq \emptyset \}$.
A space is connected if and only if it is connected as a set in the sense of belonging to one of the sets in the union of above.
Generally speaking, any non-empty subset of a topological space is connected as a set. So a set can be connected as a set as well, e.g., $\mathbb{R}$ or $\{1,2,3,4\}$.

A:

For every $A\subset X$, you can define the subspace topology as $T_A=\{U\subset X\,\vert\,A\subset U\}$. Then, $A$ is connected if and only if the subspace topology on $A$ is equal to the topology $\tau$ of $X$.

A:

The following is a more general form of the “generic answer” already given:

Any topological space is connected as a set.

I will give the proof below, but there are some other approaches using some combination of the basic points in the other answers. The proof below will still work. If you want, you could change the beginning proof in this way:

Suppose $A\subset X$ is a non-empty topological space. If \$

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