By McGraw-Hill Education

NoteablesTM: Interactive research pc with FoldablesTM is a whole note-taking process with guided notice taking for each lesson - in a workbook structure. Plus, vocabulary builder and a bankruptcy try coaching part are integrated for each bankruptcy.

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**Extra resources for Math Connects: Concepts, Skills, and Problems Solving, Course 1, Noteables: Interactive Study Notebook with Foldables (Math Connects: Course 1)**

**Example text**

Then G/H is a discrete space il and only il H is open in G. 11 H is closed, then G/H is a regular and hence Hausdortf space. 11 G/H is a 1'0 space, then H is closed and G/H is a regular space. Proof. If H is open in G, then aH is open in G for all aEG, and so cp-l({aH})=aH is open in G for every point aHEG/H. That is, every point of G/H is an open set, and hence every subset of G/H is open. Conversely, if G/H is discrete, then the set {H} is open in G/H, and thus cp-l({H}) =H is open in G. Now suppose that His closed in G.

We permit ourselves the slight soleeism of ealling such groups topologieally isomorphie. 27) is that GjH is topologieally isomorphie to G if and only if the homomorphism I earrying G onto Gis eontinuous and open [H = 1-1 (e) J. Thus G ean be reeonstructed not only as a group but also as a topologie al spaee from G and the kernel 1-1 (e) provided that I is an open, eontinuous homomorphism. Simple examples show that these restrietions on I areneeded. lJ (R), and let I be the identity mapping t of R onto itself.

That is, ab=ba for all aEA-, bEB-. 3) Corollary. 11 H is a subsemigroup, subgroup, or normal subgroup 01 a topologieal group G, then H- is also a subsemigroup, subgroup, or normal subgroup, respeetively, 01 G. 11 G is a To topologieal group and H is an Abelian subsemigroup or subgroup 01 G, then H- is also an Abelian subsemigroup or subgroup, respectively, ot G. Proof. i), we have (H-)2C (H2)-cH-; that is, H- is a subsemigroup of G. If H is a subgroup of G, then also H-1 eH. Hence (H-) -1= (H-l)- c H-, so that H- is a subgroup of G.