1. (a) What is a topological space?

(b) What is the discrete topology on a set?

Let X be a finite set and 𝒯 a topology on X for which all singleton sets {x}, x ∈ X, are closed. Show that 𝒯 is the discrete topology. Does the same conclusion hold if X is countable? Give a proof or find a counterexample.

(c) Let A be a subset of a topological space X.

What is the closure \overline{A} of A? Show that A is closed if and only if A = \overline{A}. Show that a map f : X → Y of topological spaces is continuous if and only if, for any set A ⊂ X, we have

f(\overline{A}) ⊂ \overline{f(A)}

2. (a) Let X and Y be topological spaces. What is the product topology on X × Y? Show that the product topology is the unique topology on X × Y with the property that, for any map h = (f, g) : Z → X × Y of a topological space Z, h is continuous if and only if f : Z → X and g : Z → Y are continuous.

(b) What does it mean to say that a topological space is Hausdorff ? Let f : X → Y be a map of topological spaces. The graph Γ(f) of f is given by

Γ(f) = {(x, y) ∈ X × Y : y = f(x)} ⊂ X × Y.

Let Y be Hausdorff and f continuous. Show that Γ(f) is a closed subset of X × Y with respect to the product topology. Conversely, suppose that Γ(f) is closed. Does it follow that f is continuous? Give a proof or provide a counter-example.

MATH0055 continued 3. (a) What does it mean to say that a topological space is

(i) compact? (ii) normal?

Show that a closed subset of a compact topological space is compact. Show that a compact Hausdorff topological space is normal.

(b) What does it mean to say that a topological space is connected? Let X, Y be connected topological spaces and A ⊊ X, B ⊊ Y proper subsets. Prove that X × Y \ A × B is connected. [You may assume without proof that X is homeomorphic to any X × {y}, y ∈ Y, and similarly for Y.]

4. (a) Explain why T^{2}#ℝP^{2} ≅ ℝP^{2}#ℝP^{2}#ℝP^{2}.

(b) Which surface is determined by the following triangulation?

124 235 346 457 561 672
713 134 245 356 467 571
126 237

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