### An equivalent condition for a pseudo $(k_0, k_1)$-covering space

#### Abstract

The paper aims at developing the most simplified axiom for a pseudo $(k_0, k_1)$-covering space.

To make this a success, we need to strongly investigate some properties of a weakly local ($WL$-, for short) $(k_0, k_1)$-isomorphism.

More precisely, we initially prove that a digitally topological imbedding {\it w.r.t.} a $(k_0, k_1)$-isomorphism implies a $WL$-$(k_0, k_1)$-isomorphism.

Besides, while a $WL$-$(k_0, k_1)$-isomorphism is proved to be a $(k_0, k_1)$-continuous map, it need not be a surjection. However, the converse does not hold.

Taking this approach, we prove that a $WL$-$(k_0, k_1)$-isomorphic surjection is equivalent to a pseudo-$(k_0, k_1)$-covering map, which simplifies the earlier axiom for a pseudo $(k_0, k_1)$-covering space by using one condition.

Finally, we further explore some properties of a pseudo $(k_0, k_1)$-covering space regarding lifting properties.

The present paper only deals with $k$-connected digital images.

To make this a success, we need to strongly investigate some properties of a weakly local ($WL$-, for short) $(k_0, k_1)$-isomorphism.

More precisely, we initially prove that a digitally topological imbedding {\it w.r.t.} a $(k_0, k_1)$-isomorphism implies a $WL$-$(k_0, k_1)$-isomorphism.

Besides, while a $WL$-$(k_0, k_1)$-isomorphism is proved to be a $(k_0, k_1)$-continuous map, it need not be a surjection. However, the converse does not hold.

Taking this approach, we prove that a $WL$-$(k_0, k_1)$-isomorphic surjection is equivalent to a pseudo-$(k_0, k_1)$-covering map, which simplifies the earlier axiom for a pseudo $(k_0, k_1)$-covering space by using one condition.

Finally, we further explore some properties of a pseudo $(k_0, k_1)$-covering space regarding lifting properties.

The present paper only deals with $k$-connected digital images.

### Refbacks

- There are currently no refbacks.