M6140 Topology Exercises - 5th Week (2020)
1 Topological Groups
Definition 1. A topological group (G, G, ·, 1, (−)−1) is a set G together with a topology G on G
and a group structure (·, 1, (−)−1) on G such that the multiplication ·: G × G → G and the inverse
(−)−1 : G → G are continuous. A morphism of topological groups is a continuous group homomor-
phism.
Exercise 1. Find a topological group structure on R.
Exercise 2. Find a topological group structure on C − {0}.
Exercise 3. Prove that S1 has a topological group structure.
Exercise 4. Show that GL(Rn) has a topological group structure.
Exercise 5. Prove that each topological group G is homogeneous, i.e. for each pair of points g, h ∈ G
there is a homeomorphism G → G such that g → h.
Exercise 6. Show that a topological group is T1 iff {1} is a closed set.
Exercise 7. Prove that a set U in a topological group is open iff the set U−1 := {g | g−1 ∈ U} is
open.
Exercise 8. Suppose that K1, K2 are compact subspaces of a topological group. Show that the set
K1 · K2 := {g · h | g ∈ K1, h ∈ K2} is a compact subspace too.
Exercise 9. Show that an open subgroup of a topological group is clopen.
Exercise 10. Prove that a topological group is T1 iff it is T2.
Exercise 11. Prove that the closure of a subgroup of a topological group G is a subgroup of G.
Exercise 12. Show that in a topological group every neighborhood U of 1 contains an open neighborhood
V of 1 such that V · V ⊆ U and V = V −1.
Exercise 13. Prove that a topological group is T1 iff it is T31
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