Topology-Ii

(i) A topological space X is said to be a T₃-space if it is :

(a) Regular and T₁

(b) Normal and T₁

(c) Regular and Normal

(d) T₂ and T₁

(ii) All metric spaces are :

(a) T₁ and T₂

(b) T₂ and T₃

(c) T₁, T₂, T₃ and T₄

(d) T₃ and T₁

(iii) Every closed and bounded subspace of the real line is :

(a) Connected

(b) Compact

(c) Locally Compact

(d) None of the above

(iv) Define Locally compact metric space.

(v) All projection maps are :

(a) Continuous maps

(b) Open maps

(c) Both Open and Continuous Maps

(d) None of the above

(vi) The product of any non-empty class of connected spaces is :

(a) Hausdorff

(b) Countable

(c) Connected

(d) None of the above

(vii) Define directed set and net.

(viii) Define filter on a non-empty set X.

(ix) Define null homotopy.

(x) Define homotopy of paths.

A topological space is a T₁-space ⇔ each point is a closed set.

Define the following :

(a) Hausdorff space

(b) Normal space

(c) Regular space

Prove that any closed subspace of a compact space is compact.

Prove that a topological space is compact if every basic open cover has a finite subcover.

Define product topology in terms of standard sub-base.

Prove that the product of any non-empty class of compact spaces is compact.

Define net associated with filter and filter associated with net.

Let A be a subset of a space X and let x ∈ X. Then prove that x ∈ Ā if and only if there exists a net in A which converges to x in X.

Prove that the relation homotopy (≅) and path homotopy (≅_p) are equivalence relations.

Prove that in a simply connected space X, any two paths having the same initial and final points are path homotopic.

Prove that in a Hausdorff space, any point and disjoint compact subspace can be separated by open sets in the sense that they have disjoint neighborhoods.

Prove that every compact Hausdorff space is normal.

Prove that every sequentially compact metric space is compact.

Prove that every closed and bounded subspace of the real line is compact.

Prove that the product of any non-empty class of Hausdorff spaces is Hausdorff.

Prove that the product of any non-empty class of connected spaces is connected.

Prove that a topological space is Hausdorff if and only if no filter can converge to more than one point in it.

Let T be a filter in a space X and S be the associated net in X. Then prove that x is a cluster point of the filter T if and only if it is a cluster point of the net S.

State and prove the fundamental theorem of Algebra.

Let P: E → B be a covering map and let P(e₀) = b₀. If E is path connected then prove that the lifting correspondence Φ: π₁ (B, b₀) → P^-1(b₀) is surjective. Also prove that if E is simply connected then it is bijective.