Quotient spaces
... Note that the definition implies that q must be continuous and surjective, and that the equivalence relation ∼ on X must be the one induced by q, namely x ∼ x0 if and only if q(x) = q(x0 ). How to recognize quotient maps? In sets, a quotient map is the same as a surjection. However, in topological s ...
... Note that the definition implies that q must be continuous and surjective, and that the equivalence relation ∼ on X must be the one induced by q, namely x ∼ x0 if and only if q(x) = q(x0 ). How to recognize quotient maps? In sets, a quotient map is the same as a surjection. However, in topological s ...
Problems for the exam
... 1. Let p ∈ CP2 and q ∈ RP3 . Is there a compact surface which is homotopy equivalent to CP2 \ {p} ? Is there a compact surface which is homotopy equivalent to RP3 \ {q} ? 2. Does the Borsuk-Ulam Theorem hold for the torus? In other words, is it true that for every continuous map f : S 1 × S 1 → R2 , ...
... 1. Let p ∈ CP2 and q ∈ RP3 . Is there a compact surface which is homotopy equivalent to CP2 \ {p} ? Is there a compact surface which is homotopy equivalent to RP3 \ {q} ? 2. Does the Borsuk-Ulam Theorem hold for the torus? In other words, is it true that for every continuous map f : S 1 × S 1 → R2 , ...
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... CorollaryS3.18. Let {Uλ } be an arbitrary collection of open subsets of the continuum. Then the union λ Uλ is open. Let X1 , . . . , Xn be a finite collection of closed subsets of the continuum. Then the union X1 ∪ · · · ∪ Xn is closed. Theorem 3.9 and Corollary 3.18 say that the collection T of op ...
... CorollaryS3.18. Let {Uλ } be an arbitrary collection of open subsets of the continuum. Then the union λ Uλ is open. Let X1 , . . . , Xn be a finite collection of closed subsets of the continuum. Then the union X1 ∪ · · · ∪ Xn is closed. Theorem 3.9 and Corollary 3.18 say that the collection T of op ...
Section 13. Basis for a Topology - Faculty
... Note. In this section, we consider a basis for a topology on a set which is, in a sense, analogous to the basis for a vector space. Whereas a basis for a vector space is a set of vectors which (efficiently; i.e., linearly independently) generates the whole space through the process of raking linear ...
... Note. In this section, we consider a basis for a topology on a set which is, in a sense, analogous to the basis for a vector space. Whereas a basis for a vector space is a set of vectors which (efficiently; i.e., linearly independently) generates the whole space through the process of raking linear ...
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... For example, the locus described by Y − X 2 = 0 as a subset of C2 is an affine variety over the complex numbers. But the locus described by Y X = 0 is not (as it is the union of the loci X = 0 and Y = 0). One can define a subset of affine space k n or an affine variety in k n to be closed if it is a ...
... For example, the locus described by Y − X 2 = 0 as a subset of C2 is an affine variety over the complex numbers. But the locus described by Y X = 0 is not (as it is the union of the loci X = 0 and Y = 0). One can define a subset of affine space k n or an affine variety in k n to be closed if it is a ...
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... A topological space X is compact if, for every collection {Ui }i∈I of open sets in X whose union is X, there exists a finite subcollection {Uij }nj=1 whose union is also X. A subset Y of a topological space X is said to be compact if Y with its subspace topology is a compact topological space. Note: ...
... A topological space X is compact if, for every collection {Ui }i∈I of open sets in X whose union is X, there exists a finite subcollection {Uij }nj=1 whose union is also X. A subset Y of a topological space X is said to be compact if Y with its subspace topology is a compact topological space. Note: ...
Solve EACH of the exercises 1-3
... Ex. 2. Show that a continuous image of a separable space is separable, that is, if there exists a continuous function from a separable topological space X onto a topological space Y , then Y is separable. Include the definition of a separable topological space. Ex. 3. Let f be a continuous function ...
... Ex. 2. Show that a continuous image of a separable space is separable, that is, if there exists a continuous function from a separable topological space X onto a topological space Y , then Y is separable. Include the definition of a separable topological space. Ex. 3. Let f be a continuous function ...
Lecture 4: examples of topological spaces, coarser and finer
... We claim that set of open discs forms a basis for a topology on R2 . I won’t give a rigorous proof of this, but I’ll give an illustrative diagram. Note that unlike open intervals in R, the intersection of two open discs is not an open disc. Example 7. An open rectangle in R2 is what you think it is: ...
... We claim that set of open discs forms a basis for a topology on R2 . I won’t give a rigorous proof of this, but I’ll give an illustrative diagram. Note that unlike open intervals in R, the intersection of two open discs is not an open disc. Example 7. An open rectangle in R2 is what you think it is: ...