X be a topology space show that H is open iff no limit point of His in H. 5/a/ Let (X,T) be a first countable space show that there exists a monotone decreasing local base at every point of X. of first countable space is hereditary property.

Please show steps clearly A, B and C

Let R` be the set of real numbers with the lower limit topology(the collection of the intervals [a, b) is the basis).(a) Show that the interval (1, 6) is open in R`. (b) What can you say about the interval (1, 6]? Justify your answer. (c) What about the interval (−∞, 3)?

Let R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f be 1-1? Either prove that it is or find a counterexample (a) Must g be 1-1? Either prove that it is or find a counterexample (b)

True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with the euclidean topology. The open intervals (0, 1) and (2, 3) are homeomorphic.

Can you show an example of Null space?

Q/prove:- Let (X, Td) be aspace. show that if a ex, then {a} is closed set (every finit set is closed set).

write in full sen

When ever one Point sets in X are closed a collection of functions which separates Points from closed set will separates Point. 18 (prod) is product topological space then xe A (xx, Tx) is homeomorphic to sub space of the Product space (TXA, prod). KeA The Bin Projection map 18: Tx XP is continuous and open but heed hot to be closed. Acale ctioneA} of continuos function ona topogical Space X se partes Points from closed sets inx iff the set (v) for KEA and Vopen set inx from a base for top on X-

For every element r in A, there is an element y in B such that (x, y) is in F. Can you please explain why this is differ from a relation? No proof please ! Just a couple sentences of answers!!