A Topological space is connected if it has no separation

This is equivalent to the condition that the only sets which are both open and closed are trivial (Clopen sets). Clear from the definitions.

Example: Real line with the Lower limit topology ($\mathbb{R}_l$ is not connected

Reals with Standard topology is connected.

Lemma: If $X=C\cup D$ is a separation and $Y \subset X$ (with Subspace topology) is connected, then wither $Y \subset C$ or $Y \subset D$. Proof

Lemma a union of connected subsets is conncted if they have at least one point in common. Suppose $A_\alpha \subset X$ are connected, $\alpha \in J$ and $\cap_{\alpha \in J} A_\alpha \neq \emptyset$, then their union is connected. Proof

Proposition. Let $A \subset X$ be connected, and $A \subset B \subset \bar{A}$, then $B$ is connected. In particular, the closure of a connected subspace is connected (the case $B = \bar{A}$). Proof.

Lemma. The image of a connected set by a continuous function is connected. If $f: X \rightarrow Y$ is continuous and $X$ is connected, then $f(X)$ is connected. Proof

Theorem. A product $\prod_{\alpha \in J} X_\alpha$ of connected spaces with the Product topology is connected. Proof

Linear continuum