Section 23: Connected Spaces

• Connectedness is a topological property: any two homeomorphic topological spaces are either both connected, or both disconnected, and the same set can be connected in one topology but disconnected in another, for example, $\mathbb{R}$ and $\mathbb{R}_{l}$ .
• A space is connected iff the only sets that are both open and closed in it are the whole space and the empty set.
• If a set is connected in a finer topology then it is connected in a coarser topology.

Subspaces and connectedness

• A connected subspace of a disconnected space must lie entirely within one component of any separation.
• If a connected subspace have a common point with every set in a collection of connected subspaces then the union of the set with all sets of the collection is connected.
  • If a collection of connected subspaces have a point in common then their union is connected.
• If any two consequent sets of a countable sequence of connected subspaces have a point in common then their union is connected.
• If a subspace is connected then adding some of its limit points keeps it connected.
  • This is useful, for example, for proving that the product space of any collection of connected spaces is connected in the product topology.
• If $Y\subset X$ is connected, and $Y$ intersects $A$ and $X-A$ , then $Y$ intersects $\mbox{Bd}A$ .
• If $X$ is connected, $Y\subset X$ is connected, and $(A,B)$ is a separation of $X-Y$ , then $Y\cup A$ and $Y\cup B$ are connected.

• Examples: a space with at least two points in the discrete topology; $\mathbb{Q}$ etc.

Continuous functions and connectedness

• The image of a connected space under a continuous function is connected.
• If $p:X\rightarrow Y$ is a quotient map, and $Y$ and $p^{-1}(\{y\})$ are all connected, then $X$ is connected.

Products and connectedness

• A finite product of connected spaces is connected.
  For two spaces $X$ and $Y$ : for a fixed point $y$ in $Y$ : $X\times\{y\}$ is connected; for every $x$ in $X$ : $\{x\}\times Y$ is connected.
• An arbitrary product of connected spaces is connected in the product topology.
  Fix a point in every space; every product of a finite number of spaces and fixed points for all other spaces is connected; the union of all such products is connected; the closure of the union is the whole space.
• $\mathbb{R}^{\omega}$ is connected in the product topology but not connected in the box or uniform topology.
• If $X$ and $Y$ are connected, and $A\subset X$ , $B\subset Y$ , then $X\times Y-A\times B$ is connected.