MTH634 Assignment 2 Solution 2021

MTH634 Assignment 2 Solution 2021 - VU Answer

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MTH634 ASSIGNMENT 2 SOLUTION SPRING 2021
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Question No 1:

Let X, Y be topological spaces, and let 𝑓: 𝑋 → 𝑌 be a function. Show that f is continuous if and only if for every closed set 𝐶 ⊂ 𝑌, the set f-1(𝐶) ⊂ 𝑋 is closed.

Solution:

Suppose

𝑓: 𝑋 → 𝑌 is the continuous as given

And c is closed in Y

(Y- c) is open set in Y

f-1(Y- c) is also open set in X

Result:

𝑓: 𝑋 → 𝑌 is continuous iff f-1(v) is open set in X ∀ open set v in Y so,

f-1(Y ) - f-1(c) Is open set in X

X - f-1(c) is open set in X

f-1(c) is closed in X.

Conversely

f-1(c) is closed in X ∀ closed set c in Y

Let v be an open set in Y

(Y-v) is closed set in X

f-1(Y-v) is closed set in Y

f-1(Y) - f-1(v) is closed set in X

f-1(v) is open set in X

By the above result

The function f is continuous

Hence proved

Question No 2:

Let X = {a, b, c, d} be a set and T = {∅, X, {a}, {b}, {a, b}, {c, d}, {a, c, d}, {b, c, d}} be a topology defined on X. Show that (X, T) is the first countable space.

Solution:

The given topological space is 1st countable space

Reason:

Local base at x = a

Ba={a}

Local base at x = b

Bb={b}

Local base at x = c

Bc={c,d}

Local base at x = d

Bd={c,d}

Also, every finite set with any topology is first countable.