Problem regarding the continuous real function

Reference no: EM13990716

1. A topological space (S, T ) is called perfectly normal iff for every closed set F there is a continuous real function f on S with F = f -1({0}).

(a) Show that every metric space is perfectly normal.

(b) Show that every perfectly normal space is normal. Hint: Adapt the proof of Theorem 2.6.1.

2. Let (S, T ) be a perfectly normal space and F a closed subset of S. Here is another way to prove that (a) implies (b) in Theorem 2.6.4 in such spaces. Show that there is an h ∈ Cb (S, T ) such that for every continuous function f from F into (-1, 1) and continuous function g from S into [-1, 1] which equals f on F , hg is continuous from S into (-1, 1) and equals f on F. Hint: Make h = 1 on F, 0 ≤ h <>1 elsewhere.

Reference no: EM13990716

Questions Cloud

Separated uniformity never converges : Show that if S has more than one point, a separated uniformity never converges as a filter for its product topology.

Which competitive priorities are focused on by lululemon : Which competitive priorities are focused on by lululemon

Different uniformities giving the topology : For the real line with usual topology, find two different uniformities giving the topology. Hint: Give two metrics for the topology such that the identity is not uniformly continuous from one metric to the other.

What is the density of the stretched string : An apparatus similar to the one used in the oscillation part of this experiment is used to find the speed of a wave on a string. First, we need to find the stretched string density and its uncertainty. What is the uncertainty in the density found i..

Problem regarding the continuous real function : 1. A topological space (S, T ) is called perfectly normal iff for every closed set F there is a continuous real function f on S with F = f -1({0}).

Chinese and english both have interrogative pronouns : Chinese and English both have interrogative pronouns. For this exercise, we focus on 'what' and '??/??' in English and Chinese respectively. What is interesting is that both forms can have uses that are different from the standard use of an interroga..

What factor does this increase her maximum speed : A child is on a swing, gently swinging back and forth with a maximum angle of 3.0 degree. A friend gives her a small push so that she now swings with a maximum angle of 4.0 degree. By what factor does this increase her maximum speed? Express your a..

What is the pressure in the vapor space of the evaporator : An evaporator is used to concentrate 20 wt % NaOH solution entering at 150°F to a 45 wt% NaOH solution exiting at 200°F. What is the pressure in the vapor space of the evaporator

What is the free-fall acceleration on planet x : On planet X, an object weighs 8.63 N. On planet B where the magnitude of the free-fall acceleration is 1.8 q (where g = 9.8 m/s2 is the gravitational acceleration on Earth), the object, weighs 23.03 N. What is the mass of the object on planet X? Answ..

User Account

All Pages