+1-415-670-9189
info@expertsmind.com
Finite complement topology
Course:- Mathematics
Reference No.:- EM13891864




Assignment Help
Assignment Help >> Mathematics

X and Y are sets. ζu refers to the usual topology, ζH is the half-open interval topology. ζC is the open half-line topology, ζF is the finite complement topology and ζcc. is the countable complement topology. R is the set of real numbers, Z is the set of integers and Q is the set of rational numbers.

1. If A and B are sets, then A ⊆ B if and only if A ∩ B = A.

2. Suppose that A and B are subsets of X and C and D are subsets of Y.

Then (A x C) (B x D) = (A - B) x (C - D).

3. Suppose that A and B are subsets of X and C and D are subsets of Y.

Then (A x C) U (B x D) ⊆ (A U B) x (C U D).

4. Letf : X → Y be a function and suppose that A is a subset of Y.

Then f(X- f-1 (Y - A)) ⊆ A.

5. Letf : X Y be a function and suppose that A is a subset of Y.

Then A ⊆ f(X- f-1(Y - A)) .

6. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then Bd(A) ⊆ Bd(B).

7. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then Int(A)  ⊆ Int(B).

8. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then C1(A) ⊆ Cl (B).

9. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then A' ⊆ B'

10. Suppose that (X, ζx) and (Y, ζy) are topological spaces and f : X → 0 Y is a function. If f is continuous then, for every closed set A ⊆ Y, f-1 (A) is closed.

11. Suppose that (X, ζx) and (Y, ζy) are topological spaces and f : X → Y is a function. If for every closed set A  Y, f 1(A) is closed then f is continuous.

12. Suppose that (X, ζx) and (Y, 3y) are topological spaces andf : X → Y is a function . If f is continuous then, for every closed set A ⊆ X, f(A) is closed.

13. Suppose that (X, ζx) and (Y, 3y) are topological spaces andf : X → Y is a function . If, for every closed set A ⊆ X, f(A) is closed then f is continuous.

14. Let ζ = {(-∞ , n) : n ∈ U {R, Ø}. Then ζ is a topology on R.

15. Let ζ = {(-∞,x) : x  ∈ Q} U {R, Ø} . Then ζ is a topology on R.




Put your comment
 
Minimize


Ask Question & Get Answers from Experts
Browse some more (Mathematics) Materials
A tangential-release yo-yo despin device is also included in the satellite. If the two yo-yo masses are each 7 kg, what cord length is required to completely despin the sate
Include a link to a current (or recent) ethical dilemma and discuss your thoughts on the dilemma, your perceptions of your co-worker's response, and possible resolutions wit
Find the intervals on which f is increasing. Enter the interval that contains smaller numbers first. Find the interval on which f is decreasing. Find the local minimum and max
Which normal distribution has a wider spread: Distribution A with a mean of one and a standard deviation of two, or Distribution B with a mean of two and a standard deviatio
Explain the theory developed including capacitated s,t graphs and the lexicographic ordering rule based on the case study and suggested readings. Give an example of how this
An anchor weighing 105 lb in water is attached to a chain weighing 2 lb/ft in water. Find the work done to haul the anchor and chain to the surface of the water from a depth
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or "washe
The future value of a simple interested investment is given by S=P(1+rt), where P is the principal invested at a simple interest rate r for t years. What principal P must b