Connected spaces: Tutorial problems
1. Show that the set B = {(x, y) ∈ R2 : x2 + y 2 < 1} is connected.
2. Consider the hyperbola H = {(x, y) ∈ R2 : xy = 1}. How many
connected components does it have? Explain.
3. Let X = {0, 1, 2} and let T = {∅, X, {0, 1}, {1}, {1, 2}}. Is the
topological space (X, T ) connected? Is it Hausdorff?
4. Find a set U ⊂ R and a differentiable function f : U → R whose
derivative f ′ is identically zero, even though f is not constant.
5. Find a connected topological space whose topology is discrete.
6. Suppose A, B are open subsets of X such that A ∩ B, A ∪ B are
both connected. Show that A, B must be connected as well.
Connected spaces: Some hints
1. Define f : (0, 1) × [0, 2π) → R2 by f (r, θ) = (r cos θ, r sin θ).
2. Define f : (0, ∞) → R2 by f (x) = (x, 1/x). Then f is continuous
and its image is one of the two connected components.
3. It is connected, but it is not Hausdorff.
4. Let U = (0, 1) ∪ (1, 2) and let f (x) =
1 if 0 < x < 1
2 if 1 < x < 2
.
5. Such a topological space can have at most one element.
6. Suppose U, V form a partition of A. Since A ∩ B is connected, one
may assume that A ∩ B ⊂ U without loss of generality. Show that the
sets V and U ∪ B form a partition of A ∪ B. This is a contradiction.