Let $A$ be connected and $U$ be clopen in $Y-A$. Prove $A\cup U$ is connected. [Dugundji]
62911
Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
18572
Union of a family of connected pairwise not disjoint subspaces is connected
86505
Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
102924
Normal space- equivalent condition: $\forall U, V$ - open $: U \cup V = X \ \ \ \exists F \subset U \ , G \subset V$ - closed $: F \cup G = X$
89264
Connectedness in the Quotient Space
124234
Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
158850
Is it true that a boundary of a simply connected and bounded set is connected in $\mathbb{C}$?
45720
Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$
109046
Is the complement of an open ball in a Banach space connected?
115606
Proofs involving $ (A\setminus B) \cup (A \cap B)$
60707
Show that if $ \ A \cup B = A$ and $ \ A\cap B = A$ then $ \ A = B$
48557
Product of connected spaces
35069
Connectedness of the boundary
90619
For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
70959
Prove that a set $E$ is closed iff it's complement $E^{c}$ is open
152193
The intersection of a closure with an open set
143735
Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$