To show that the only nonempty subset of $\Bbb R$ which is both open and closed in $\Bbb R$ is $\Bbb R$.
2575
Showing that $\mathbb{R}$ is connected
42494
The boundary is a closed set
13066
A Zariski open subset of a variety has the same dimension as the variety.
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
82825
Proving set of interiors is open
91991
How do I prove that two closed sets are separable?
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
146146
Proof that a closed subset of a compact space is a compact space
13848
Showing that if a subset of a complete metric space is closed, it is also complete
50880
Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
16654
Unions and Intersections of Open Sets are Open
146664
The union of two linearly independent subsets
70264
Prove that the inverse image of an open set is open
123149
Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
158569
Proving a particular subset of $R^n$ is closed
21599
$(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
127173
Closed subset of compact set is compact