Sufficient condition for the density

In summary, the space of all continuous real-valued functions on a completely Hausdorff space X with the topology of pointwise convergence will be dense in the Tychonoff product ℝX, as the Tychonoff product topology is the strongest topology and the space of continuous real-valued functions is dense in this topology. This means that every point in ℝX has a "neighborhood" of continuous real-valued functions surrounding it.
  • #1
Flem
2
0
I'm sorry for my bad English.
Let X a completely Hausdorff space. Will the be space of all continuous real-valued functions on a space X with the topology of pointwise convergence dense in the Tychonoff product ℝX?
 
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  • #2
I'm sorry for my foolish. There is nothing difficult. The question is closed.
 
  • #3


Hi there! Don't worry about your English, it's perfectly understandable. To answer your question, yes, the space of all continuous real-valued functions on a completely Hausdorff space X with the topology of pointwise convergence will be dense in the Tychonoff product ℝX. This is because the Tychonoff product topology is the strongest topology on ℝX, and the space of continuous real-valued functions is dense in this topology. So, you can think of it as every point in the Tychonoff product ℝX having a "neighborhood" of continuous real-valued functions around it. I hope this helps!
 

Related to Sufficient condition for the density

1. What is a sufficient condition for the density of a substance?

A sufficient condition for the density of a substance is a measure of its mass per unit volume. It is typically expressed in units of grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

2. How is density related to the mass and volume of a substance?

Density is directly related to the mass and volume of a substance. As the mass of a substance increases, its density also increases. Similarly, as the volume of a substance increases, its density decreases.

3. Can the density of a substance change?

Yes, the density of a substance can change under different conditions such as temperature and pressure. For example, as the temperature of a substance increases, its particles have more energy and spread out, resulting in a decrease in density.

4. What is the importance of knowing the density of a substance?

Knowing the density of a substance is important for a variety of reasons. It can help determine the purity or composition of a substance, as well as its suitability for certain applications. For example, the density of a material can affect its buoyancy in water or its ability to conduct electricity.

5. How is density measured experimentally?

Density can be measured experimentally by measuring the mass and volume of a substance and then dividing the mass by the volume. This can be done using various techniques such as displacement, liquid displacement, and gravimetric analysis.

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