Union of path connected topological spaces again path connected?

Question

In our topology script there was the following remark after a couple theorems about path connectedness:

If $(A_j,j \in J)$ a family of path connected spaces, all sharing a point $x$. Then $\bigcup_{j \in J}A_j$ is a path connected space.

My question now is:

• How is the union of topological spaces again a topological space?
• Should $A_j$ not be subsets of a bigger topological space $E$ to say that the union is a topological space? Is then the new topology on $\bigcup_{j \in J}A_j$ the induced topology on this set?

Answer 1

Indeed, implied in the statement is really:

Suppose $X$ is a space and $A_j, j \in J$ is a family of subsets of $X$ with a common point $x$. Suppose also that each $A_j$ in the subspace topology w.r.t. $X$ is a path-connected space.
Then $\cup_j A_j$ in the subspace topology w.r.t. $X$ is also path-connected.

But this is quite verbose so often abbreviated as in your notes.