Week 6
Define a topological space X with a subspace A. Find and describe a pair of sets that are a separation of A in X. In addition look at your classmates’ topological spaces, subspaces, and separated sets. Discuss the validity of their separation.
For example: Consider ℝu, ℝ with the upper limit topology, whose basis elements are (a,b] where a < b. Let A = [1,2] so A ⊂ ℝ. Define U = (0,1] and V = (1,3] and let A' = A ⋂ U and A'' = A ⋂ V. Then A' and A'' are open in the subspace topology for A since U and V are open in ℝu. Note that:
A = A' ⋃ A'' A' = {1} A'' = (1,2] A' ⋂ A'' = ∅
Thus, A' and A'' are non-empty disjoint open (in the subspace topology for A) sets whose union is A. Thus, A' and A'' are a separation of A in ℝu and A is a disconnected subspace of ℝu. Note that no separation of A exists in ℝ using the standard topology.
Why Choose Us
- 100% non-plagiarized Papers
- 24/7 /365 Service Available
- Affordable Prices
- Any Paper, Urgency, and Subject
- Will complete your papers in 6 hours
- On-time Delivery
- Money-back and Privacy guarantees
- Unlimited Amendments upon request
- Satisfaction guarantee
How it Works
- Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
- Fill in your paper’s requirements in the "PAPER DETAILS" section.
- Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
- Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
- From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.