If D is dense in a topological space (X, tau), prove that D is dense .pdf

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If D is dense in a topological space (X, tau), prove that D is dense in every topology smaller than tau. Solution If (X,T) is atopological space, then X is the set and T is the topology on X.T is also the family of subsets of X. If an element in the set X, D refers to \'dense\', then, an element D in T refers to dense only. A subset can derive elements only from its superset..

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Define nonconcordant variation: Why does nonconcordant variation make creating ‘race’ categories difficult? Why does clinal variation make creating ‘race’ categories difficult? Describe how genetic variation is distributed between and within human populations: Why does this distribution make creating ‘race’ categories difficult? Solution Part I If two pairs of genetically controlled traits are taken, then it is found that the two traits are not correlated to each other, for e.g. height and skin color are not at all related. Such type of traits is called as Non-concordant variation. So, one cannot predict the flow of traits from one race to another. Race categories are always defined on the basis of genetically related traits and variations between different traits. Now if there is no correlation, then defining of races will become difficult. For e.g. a tall man as well a short man or an average heighted man; all can have extremely fair complexion. So, you cannot associate the races of fair complexion, say Americans, with height. It is not compulsory that all tall people are fair, or all short people are dark. Part II Clinal variation: This is a change related to environmental variation like temperature change, altitude change, and change in levels of precipitation. If there is a gradual change in all populations with a particular environmental variation, then the type of variation is called as ‘clinal’ variation. One e.g. is number of counter folds in molar teeth. These changed gradually in all populations. So, you cannot distinguish one race from another on the basis of a clinal change. ‘Clinal’ change is a change which has occurred in all existing populations. Part III Genetic variation in humans is very diverse. The variations can occur due to single nucleotide changes (SNPs) or changes in multiple nucleotides. Let\'s talk about SNPs. There may be two copies of a single nucleotide at a particular position in one human being. On the other hand, there may multiple copies of the same nucleotide at the same position in another human being. I mean to say, in multiple human beings, multiple type of SNPs are found, all related to number of copies of the single nucleotide; creating variations among all species. There is no specific pattern of formation of SNP in a particular race, class or tribe. So, on basis of these SNPs, you will never be able to classify different races. Overall, there is no single formula to classify a race. Therefore, creating \'race\' categories is difficult..

Define nonconcordant variation Why does nonconcordant variation mak.pdf

If D is dense in a topological space (X, tau), prove that D is dense .pdf

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If D is dense in a topological space (X, tau), prove that D is dense .pdf